..'•  •  -    vi .....    )    v. 


L 


THEORY  OF 
MAXIMA  AND  MINIMA 


BY 
HARRIS  HANCOCK,  PH.D.  (BERLIN),  DR. So.  (PARIS) 

PROFESSOR  OF  MATHEMATICS  IX  THE 
UNIVERSITY   OF  CINCINNATI 


GINN  AND  COMPANY 

BOSTON     •     NEW    YORK     •     CHICAGO     •     LONDON 
ATLANTA      •     DALLAS     •     COLUMBUS     •     SAN    FRANCISCO 


COPYRIGHT,  1917,  BY 
HARRIS  HANCOCK 


ALL    RIGHTS   RESERVED 
217.11 


tgfte   satbenaum 

GINN  AND  COMPANY  •  PRO 
PRIETORS  •  BOSTON  •  U.S.A. 


PREFACE 

Mathematicians  have  always  been  occupied  with  questions  of 
maxima  and  minima.  With  Euclid  one  of  the  simplest  problems 
of  this  character  was :  Find  the  shortest  line  which  may  be  drawn 
from  a  point  to  a  line,  and  in  the  fifth  book  of  the  conies  of 
Apollonius  of  Perga  occur  such  problems  as  the  determination 
of  the  shortest  line  which  may  be  drawn  from  a  point  to  a  given 
conic  section. 

It  is  thus  seen  that  a  sort  of  theory  of  maxima  and  minima 
was  known  long  before  the  discovery  of  the  differential  calculus, 
and  it  may  be  shown  that  the  attempts  to  develop  this  theory 
exercised  considerable  influence  upon  the  discovery  of  the  cal 
culus.  Fermat,  for  example,  after  making  numerous  restorations 
of  two  books  of  Apollonius,  often  cites  this  old  geometer  in.  his 
"  method  for  determining  maximum  a  nd  minimum"  1638,  a  work 
which  in  some  instances  is  so  closely  related  to  the  calculus 
that  Lagrange,  Laplace,  Fourier,  and  others  wished  to  consider 
Fermat  as  the  discoverer  of  the  calculus.  This  he  probably  would 
have  been  had  he  started  from  a  somewhat  more  general  point 
of  view,  as  in  fact  was  done  by  Newton  (Opuscula  Newtoni,  /, 
86-88). 

Maclaurin  (A  Treatise  of  Fluxions,  Vol.  I,  p.  214.  1742),  wrote  : 
"  There  are  hardly  any  speculations  in  geometry  more  useful  or 
more  entertaining  than  those  which  relate  to  maxima  and  minima. 
Amongst  the  various  improvements  that  began  to  appear  in  the 
higher  parts  of  geometry  about  a  hundred  years  ago,  Mr.  de 
Fermat  proposed  a  method  for  rinding  the  maxima  and  minima. 
How  the  methods  that  were  then  invented  for  the  mensuration 
of  figures  and  drawing  tangents  to  curves  are  comprehended 
and  improved  by  the  method  of  Fluxions,  may  be  understood 
from  what  has  already  been  demonstrated.  A  general  way  of 

iii 


iv  THEOEY  OF  MAXIMA  AND  MINIMA 

resolving  questions  concerning  maxima  and  minima  is  also  de 
rived  from  it,  that  is  so  easy  and  expeditious  in  the  most 
common  cases,  and  is  so  successful  when  the  question  is  of  a 
higher  degree,  when  the  difficulty  is  greater  and  other  methods 
fail  us,  that  this  is  justly  esteemed  one  of  the  most  admirable 
applications  of  Fluxions." 

The  theory  of  maxima  and  minima  was  rapidly  developed 
along  the  lines  of  the  calculus  after  the  discovery  of  the  latter. 
Mathematicians  were  at  first  satisfied  with  finding  the  necessary 
conditions  for  the  solution  of  the  problem.  These  conditions,  how 
ever,  are  seldom  at  the  same  time  sufficient.  In  order  to  decide  this 
last  point,  the  discovery  of  further  algebraic  means  was  necessary. 
Descartes  had  already  remarked,  in  a  letter  of  March  1, 1638,  that 
Fermat's  rule  for  finding  maxima  and  minima  was  imperfect ;  and 
we  shall  see  that  many  imperfections  still  existed  for  a  long  time 
after  the  invention  of  the  calculus  by  Newton. 

As  introductory  to  a  course  of  lectures  on  the  calculus  of 
variations,  I  have  for  a  number  of  years  given  a  brief  outline 
of  the  theory  of  maxima  and  minima.  This  outline  is  founded 
on  the  lectures  that  were  presented  by  the  late  Professor 
Weierstrass  in  the  University  of  Berlin.  It  treats  the  ordinary 
cases ;  that  is,  where  the  functions  are  everywhere  regular  and 
where  the  forms  are  either  definite  or  indefinite.  It  was  published 
as  a  bulletin  of  the  University  of  Cincinnati  in  1903.  At  that 
time  I  expected  to  publish  another  bulletin  which  was  to  treat 
the  more  special  cases ;  for  example,  where  only  one-sided  differ 
entiation  enters,  the  "ambiguous  case,"  where  the  form  is  semi- 
definite,  etc.  A  treatment  of  these  cases,  the  extraordinary  cases, 
required  more  study  than  was  anticipated.  The  bulletin  has 
consequently  been  delayed  so  long  that  I  have  concluded  to  give 
an  entirely  new  exposition  of  the  whole  theory. 

In  the  preface  to  the  German  translation  by  Bohlmann  and 
Schepp  of  Peano's  Calcolo  differenziale  e  principii  di  calcolo 
integrale,  Professor  A.  Mayer  writes  that  this  book  of  Peano  not 
only  is  a  model  of  precise  presentation  and  rigorous  deduction, 
whose  propitious  influence  has  been  unmistakably  felt  upon 


PREFACE  v 

almost  every  calculus  that  has  appeared  (in  Germany)  since  that 
time  (1884),  but  by  calling  attention  to  old  and  deeply  rooted 
errors  it  has  given  an  impulse  to  new  and  fruitful  development. 

The  important  objection  contained  in  this  book  (Xos.  133-136) 
showed  unquestionably  that  the  entire  former  theory  of  maxima 
and  minima  needed  a  thorough  renovation ;  and  in  the  main 
Peano's  book  is  the  original  source  of  the  beautiful  and  to  a 
great  degree  fundamental  works  of  Scheeffer,  Stolz,  Victor  v. 
Dantscher,  and  others,  who  have  developed  new  and  strenuous 
theories  for  extreme  values  of  functions.  Speaking  for  the 
Germans,  Professor  A.  Mayer,  in  the  introduction  to  the  above- 
mentioned  book,  declares  that  there  has  been  a  long-felt  need 
of  a  work  which,  for  the  first  time,  not  only  is  free  from  mis 
takes  and  inaccuracies  that  have  been  so  long  in  vogue  but 
which,  besides,  so  incisively  penetrates  an  important  field  that 
hitherto  has  been  considered  quite  elementary. 

To  a  considerable  degree  these  inaccuracies  are  due  to  one  of 
the  greatest  of  all  mathematicians,  Lagrange,  and  they  have 
been  diffused  in  the  French  school  by  Bertrand,  Serret,  and 
others.  "We  further  find  that  these  mistakes  are  ever  being 
repeated  by  English  and  American  authors  in  the  numerous 
new  works  which  are  constantly  appearing  on  the  calculus. 

It  seems,  therefore,  very  desirable  in  the  present  state  of 
mathematical  science  in  this  country  that  more  attention  be 
given  to  the  theory  of  maxima  and  minima ;  for  it  has  a  high 
interest  as  a  topic  of  pure  analysis  and  finds  immediate  appli 
cation  to  almost  every  branch  of  mathematics. 

I  have  therefore  prepared  the  present  book  for  students  who 
wish  to  take  a  more  extended  course  in  the  calculus  as  intro 
ductory  to  graduate  work  in  mathematics.  I  do  not  believe  in 
making  university  students  study  abstruse  theories  in  foreign 
languages,  and  in  this  treatise  it  will  be  found  that  the  peda 
gogical  side  of  the  presentation  is  insisted  upon;  for  example, 
the  Taylor  development  in  series  is  given  under  at  least  half  a 

dozen  different  forms. 

HARRIS  HAXCOCK 


CONTENTS 

CHAPTER  I 

FUNCTIONS  OF  ONE  VARIABLE 
I.    ORDINARY  MAXIMA  AND  MINIMA 

SECTION  PAGE 

1.  Greatest  and  smallest  value  of  a  function  in  a  fixed  interval ;  maximum 

and  minimum.  Several  maxima  or  minima  in  the  same  interval. 
Ordinary  and  extraordinary,  proper  and  improper  maxima  and  minima. 
The  extremes  of  functions;  upper  and  lower  limits;  absolute  and 
relative  maxima  and  minima 1 

2.  Criterion  for  maxima  or  minima  values 2 

3.  Other  theorems 4 

4.  Re'surne'  of  the  above  expressed  in  a  somewhat  different  form   ....       5 

II.    EXTRAORDINARY  MAXIMA  AND  MINIMA 
A.    Functions  which  have  Derivatives  only  on  Definite  Positions 

5.  Left-hand  and  right-hand  differential  quotients 6 

6.  Criteria  as  to  whether  a  root  of  the  equation /(x)  =  0  offers  an  extreme 

of  the  f unction /(x).    Statement  and  proofs  of  certain  theorems    .    .       7 

B.  Functions  which  have  only  One-Sided  Differential  Quotients  of  a 

Certain  Order  for  a  Value  x  =  XQ 

7.  Theorems  and  their  proofs 11 

C.  Upper  and  Lower  Limits  of  a   One-Valued  Function  which   is 
Continuous  for  Values  of  the  Argument  within  a  Definite  Interval 

8.  Greatest  and  least  values  ;  upper  and  lower  limits.   Examples 12 

9.  An  interesting  example  given  by  Liouville 13 

10.  Examples  in  which  the  functions  are  discontinuous  ;  cases  in  which 

the  upper  and  lower  limits  are  not  reached 14 

PROBLEMS       15 

vii 


viii  THEORY  OF  MAXIMA  AND  MINIMA 

CHAPTER  II 

FUNCTIONS  OF  SEVERAL  VARIABLES 
I.    ORDINARY  MAXIMA  AND  MINIMA 


SECTION 


Preliminary  Remarks 


PAGE 


11.  Proper  and  improper  extremes 17 

12.  Criteria  for  maxima  and  minima 17 

13.  Definite,  semi-definite,  and  indefinite  forms 19 

14.  Criterion  for  a  definite  quadratic  form .  19 

II.    RELATIVE  MAXIMA  AND  MINIMA 

15.  Statement  of  the  problem.    Derivation  of  the  conditions 21 

PROBLEMS 22 

CHAPTER  III 

FUNCTIONS  OF  TWO  VARIABLES 
I.    ORDINARY  EXTREMES 

16.  Definitions  and  derivation  of  the  required  conditions.    Statement 

of  different  cases  after  Goursat 23 

17.  The  indefinite  case 25 

18.  The  definite  case 25 

19.  Summary  of  the  results  derived  for  the  two  cases  above.    Example 

taken  from  the  theory  of  least  squares 26 

PROBLEMS 27 

Introduction  to  the  Ambiguous  Case  B2  —  A  C  =  0 

20-23.  Exposition  of  the  attending  difficulties  which  are  illustrated  by 

means  of  geometric  considerations  due  to  Goursat 27 

24.  The  classic  example  of  Peano  showing  when  we  may  and  may  not 

expect  extremes 31 

II.    INCORRECTNESS  OF  DEDUCTIONS  MADE  BY  EARLIER  AND 
MANY  MODERN  WRITERS 

25.  The  Lagrange  fallacy  followed  by  Bertrand,  Serret,  Todhunter, 

etc.    Geometric  explanation  of  this  fallacy  . 33 

III.    DIFFERENT  ATTEMPTS  TO  IMPROVE  THE  THEORY 

26.  Failure  in  deriving  the  required  conditions  by  studying  the  behavior 

of  the  function  upon  algebraic  curves 35 

27.  Statement  of  the  Scheeffer  method 37 

28.  The  method  of  Von  Dantscher  outlined 39 

29.  The  Stolzian  theorems  .    .  39 


CONTENTS  ix 

CHAPTER  IV 
THE  SCHEEFFER  THEORY 

I.    GENERAL  CRITERIA  FOR  A  GREATEST  AND  A  LEAST  VALUE  OF 
A  FUNCTION  OF  Two  VARIABLES  ;  IN  PARTICULAR  THE  EXTRAOR 
DINARY  EXTREMES 

SECTION  PAGE 

30.  Stolz's  analytic  proof  of  the  Scheeffer  theorem 43 

31.  Stolz's  added  theorem  by  which  the  nonexistence  of  extremes  may 

be  often  ascertained 45 

32.  Scheeffer' s  geometric  proof  of  his  own  theorem 46 

33.  Failure  of  this  theorem 48 

II.    HOMOGENEOUS  FUNCTIONS 

34.  Scheeffer's  criteria  of  extremes  for  such  functions 49 

III.    EXTREMES  FOR  FUNCTIONS  .THAT  ARE  NOT  HOMOGENEOUS 

35.  In  particular  the  case  where  the  terms  of  lowest  dimension  consti 

tute  a  semi-definite  form.    Extreme  curves 52 

36.  The  criteria  established 54 

37-40.  A  direct  and  more  practical  method  due  to  Stolz  of  deriving  the 

above  results 55 

41.  Exceptional  cases    .    .    .    .  • 58 

EXAMPLES 59 

IV.    THE  METHOD  OF  VICTOR  v.  DANTSCHER 

42.  The   general   method    outlined   and   applied   to   the   definite   and 

indefinite  cases 62 

43-44.  Treatment  of  the  semi-definite  case .     64 

PROBLEMS 69 

V.    FUNCTIONS  OF  THREE  VARIABLES 
Treatment  in  Particular  of  the  Semldefinite  Case 

45.  Extensions  of  the  theorems  and  proofs  given  by  Stolz  and  Scheeffer 

for  functions  of  two  variables 70 

46.  Evident  generalizations  of  §§  37-41 71 

PROBLEMS 72, 


x  THEOKY  OF  MAXIMA  AND  MINIMA 

CHAPTER  V 

MAXIMA  AND  MINIMA  OF  FUNCTIONS  OF  SEVERAL  VARIABLES 
THAT  ARE  SUBJECTED  TO  NO  SUBSIDIARY  CONDITIONS 

I.    ORDINARY  EXTREMES 

SECTION  PAGE 

47.  Nature  of  the  functions  under  consideration.   Definition  of  regular 

functions 73 

48.  Definition  of  maxima  and  minima  of  functions  of  one  and  of  several 

variables 74 

49.  The  problem   of   this  chapter  proposed.    Taylor's  theorem  for 

functions  of  one  variable 75 

50.  Taylor's  theorem  for  functions  of  several  variables 77 

51.  The  usual  form  of  the  same  theorem 79 

52.  A  condition  of  maxima  and  minima  of  such  functions 81 

II.  THEORY  OF  THE  HOMOGENEOUS  QUADRATIC  FORMS 

63.  No  maximum  or  minimum  value  of  the  function  can  enter  when 
the  corresponding  quadratic  form  is  indefinite.  When  is  a 
quadratic  form  a  definite  form  ? 82 

54.  Some  properties  of  quadratic  forms.  The  condition  that  the 
quadratic  form  <f>  (xv  x2,  •  •  •,  xn)  =  S^A^XAX^  be  expressible 

as  a  function  of  n  —  1  variables 83 

55-61.  Every  homogeneous  function  of  the  second  degree  0  (xv  x2,  -  •  -,  xn) 
may  be  expressed  as  an  aggregate  of  squares  of  linear  functions 
of  the  variables 85 

62.  The  question  of  §  53  answered 91 

III.   APPLICATION  OF  THE  THEORY  OF  QUADRATIC  FORMS  TO  THE 
PROBLEM  OF  MAXIMA  AND  MINIMA  STATED  IN  §§  47~51 

63.  Discussion  of  the  restriction  that  the  definite  quadratic  form  must 

only  vanish  when  all  the  variables  vanish.   The  problem  of  this 
chapter  completely  solved 92 

64.  Upper  and  lower  limits 94 

CHAPTER  VI 

THEORY  OF  MAXIMA  AND  MINIMA  OF  FUNCTIONS  OF  SEVERAL 

VARIABLES  THAT  ARE  SUBJECTED  TO  SUBSIDIARY  CONDITIONS. 

RELATIVE  MAXIMA  AND  MINIMA 

65.  The  problem  stated 96 

66.  The  natural  way  to  solve  it 96 

67-70.  Derivation  of  a  fundamental  condition 96 

71.  Another  method  of  finding;  the  same  condition 98 


CONTEXTS  xi 

SECTION  PAGE 

72.  Discussion  of  the  restrictions  that  have  been  made 100 

73.  A  geometrical  illustration  of  these  restrictions 101 

74.  Establishment  of  certain  criteria 102 

75.  Simplifications  that  may  be  made - 102 

76.  More  symmetric  conditions  required 103 

I.  THEORY  OF  HOMOGENEOUS  QUADRATIC  FORMS 

77.  Addition  of  a  subsidiary  condition 103 

78,  79.  Derivation  of  the  fundamental  determinant  Ae  and  the  discussion 

of  the  roots  of  the  equation  Ae  =  0,  known  as  the  equation  of 

secular  variations 104 

79.  The  roots  of  this  equation  are  all  real 107 

80-82.  Weierstrass's  proof  of  the  above  theorem 108 

83,  84.  An  important  lemma 109 

85.  A  general  proof  of  a  general  theorem  in  determinants Ill 

86.  The  theorem  proved  when  the  variables  are  subjected  to  subsidiary 

conditions 112 

87.  Conditions  that  the  quadratic  form  be  invariably  positive  or  invari 

ably  negative 114 

II.    APPLICATION  OF  THE  CRITERIA  JUST  FOUND  TO  THE 
PROBLEM  OF  THIS  CHAPTER 

88.  The  problem  as  stated  in  §  65  solved  and  formulated  in  a  symmetric 

manner 114 

89.  The  results  summarized  and  the  general  criteria  stated 115 

90.  Discussion  of  the  geometrical  problem  :  Determine  the  greatest  and 

the  smallest  curvature  at  a  regular  point  of  a  surface.    Derivation 

of  the  characteristic  differential  equation  of  minimal  surfaces     .     116 

91.  Solution  of  the  geometrical  problem  :    From  a  given  point  to  a 

given  surface  draw  a  straight  line  whose  length  is  a  minimum  or 

a  maximum 123 

92.  Brand's  problems.    Two  problems  taken  from  the  theory  of  light. 

Keflection.    Refraction 126 

CHAPTER  VII 
SPECIAL  CASES 

I.  THE  PRACTICAL  APPLICATION  OF  THE  CRITERIA  THAT  HAVE  BEEN 
HITHERTO  GIVEN  AND  A  METHOD  FOUNDED  UPON  THE  THEORY  OF 
FUNCTIONS,  WHICH  OFTEN  RENDERS  UNNECESSARY  THESE  CRITERIA 

93.  Difficulties  that  are  often  experienced.    Fallacies  by  which  maxima 

and  minima  are  established  when  no  such  maxima  or  minima 
exist  134 


xii  THEOKY  OF  MAXIMA  AND  MINIMA 

SECTION  PAGE 

94.  Definitions :  Realm.    A  position.    An  n-ple  multiplicity.    Struc 

tures.   A  position  defined  which  lies  within  the  interior  of  a 
definite  realm,  on  the  boundary,  or  without  this  realm      .    .    .     135 

95.  Statement  of  two  important  theorems  in  the  theory  of  functions    136 

96.  Upper  and  lower  limits  for  the  values  of  a  function.    Asymptotic 

approach.    Geometrical  and  graphical  illustrations 137 

97.  Digression  into  the  theory  of  functions 138 

II.  EXAMPLES  OF  IMPROPER  EXTREMES 

98.  Cases  where  there  are  an  infinite  number  of  positions  on  which 

a  function  may  have  a  maximum  value 140 

99.  Reduction  of  such  cases  to  the  theory  of  maxima  and  minima 

proper.   The  derivatives  of  the  first  order  must  vanish    .    .    .     141 

100.  The  results  that  occur  here  are  just  the  condition  which  made 

the  former  criteria  impossible 142 

101.  The  previous  investigations  illustrated  by  the  problem  :    Among 

all  polygons  which  have  a  given  number  of  sides  and  a  given 
perimeter,  find  the  one  which  contains  the  greatest  surface-area    .     142 
102-107.  Solution  of  the  above  problem 143 

102.  Cremona's  criterion  as  to  whether  a  polygon  has  been  described 

in  the  positive  or  negative  direction      143 

PROBLEM 147 

108.  A  problem  due  to  Hadamard 147 

III.    CASES  IN  WHICH  THE  SUBSIDIARY  CONDITIONS  ARE  NOT  TO 

BE    REGARDED    AS    EQUATIONS    BUT    AS    LIMITATIONS 

109.  Examples  illustrating  the  nature  of  the  problem  when  the  vari 

ables  of  a  given  function  cannot  exceed  definite  limits    .    .    .  149 

110.  Reduction  of  two  inequalities  to  one 149 

111.  Inequalities  expressed  in  the  form  of  equations 150 

112.  Examples  taken  from  mechanics 150 

GAUSS'S  PRINCIPLE 

113.  Statement  of  this  principle 151 

114.  Its  analytical  formulation .     152 

115.  By  means  of  this  principle  all  problems  of  mechanics  may  be 

reduced  to  problems  of  maxima  and  minima 152 

116.  Proof  of  a  theorem  in  the  theory  of  functions 153 


THE  REVERSION  OF  SERIK> 


117-126.  Proof  and  discussion  of  this  theorem.    Determination  of  upper 

and  lower  limits  for  the  quantities  that  occur 154 

123.  Unique  determination  of  the  system  of  values  of  the  x's  that 

satisfy  the  equations 159 


CONTENTS  xiii 

SECTION  PAGE 

124.  If  the  y's  become  infinitely  small  with  the  x's,  the  x's  become 

infinitely  small  with  the  y's.   The  x's  are  continuous  functions 

of  the  y's 160 

125.  The  x's,  considered  as  functions  of  the  y's,  have  derivatives 

which  are  continuous  functions  of  the  y's.   The  existence  of 

the  first  derivatives 162 

126.  The  x's,  expressed  in  terms  of  the  y's,  are  of  the  same  form  as 

the  given  equations  expressing  the  y's  in  terms  of  the  x's    .    .     163 

127.  Conditions  which  must  exist  before  the  ordinary  rules  of  differ 

entiation  are  allowable  in  the  most  elementary  cases  ....     163 
MISCELLANEOUS  PROBLEMS 164 

CHAPTER  VIII 

CERTAIN  FUNDAMENTAL  CONCEPTIONS  IN  THE  THEORY  OF 
ANALYTIC  FUNCTIONS 

I.    ANALYTIC  DEPENDENCE;  ALGEBRAIC  DEPENDENCE 

128.  Rational  functions  of  one  or  more  variables.    Functions  defined 

through  arithmetical  operations.  One-value  functions.  Infi 
nite  series  and  infinite  products.  Convergence  ......  166 

129.  Uniform  convergence 167 

130.  Region  of  convergence.    Differentiation 167 

131.  Functions  of  several  variables.   Functions  which  behave  like  an 

integral  rational  function  .    . 168 

132.  Analytic  dependence 169 

133.  Many-valued  functions 169 

134.  Possibility  of  expressing  many-valued  functions  through  one- 

valued  functions 169 

135.  An  important  theorem  for  the  calculus  of  variations 171 

136.  The  same  theorem  proved  in  a  more  symmetric  manner  ....     172 

137.  Application  of  this  theorem  and  the  definition  of  a  structure 

of  the  (71  —  w)th  kind  in  the  realm  of  n  quantities.  Analytic 
continuation 174 

138.  Analytic  structures  defined  in  another  manner.    Structures  of 

the  first  kind,  second  kind,  etc 175 

II.    ALGEBRAIC  STRUCTURES  IN  Two  VARIABLES 

139.  A  Igebraic  structure  defined ;  regular  point ;  singular  point    .    .    .     176 
140,  141.  Development  of  an  algebraic  function  y  in  the  neighborhood  of 

a  regular  point.  For  such  a  point  there  is  only  one  power 
series  in  x  which  when  written  for  y  makes  identically  zero 
the  function  which  defines  the  algebraic  structure 177 


xiv  THEORY  OF  MAXIMA  AND  MINIMA 

III.    METHOD   OF  FINDING  ALL   SERIES  FOR  y  WHICH  BELONG  TO 
A  k-ply  SINGULAR  POINT 

SECTION  PAGE 

142-143.  Practical  methods  for  the  derivation  of  such  series  under  the 

different  conditions  that  may  arise 180 

144.  Theorem  of  Stolz  which  serves  as  a  check  on  the  derivation  of 

the  above  series 187 

145.  An  example  illustrating  the  general  theory ]87 

INDEX  ......  ]91 


THEORY  OF  MAXIMA  AND  MINIMA 

CHAPTER  I 

FUNCTIONS  OF  ONE  VARIABLE 
I.    ORDINARY  MAXIMA  AND  MINIMA 

1 .  A  function  f(x)  which  is  uniquely  defined  for  all  values  of 
x  in  the  interval  (a,  I)  is  said*  to  have  a  greatest  value  or  a  maxi- 
mum  for  the  value  x  =  xQt  situated  between  a  and  b,  if  there  is 
a  positive  quantity  8  such  that  for  all  values  of  h  between  —  8 
and  +  8  the  difference 

[i]  /(*„+*) -/(*„>=<) 

exists,  which  at  the  same  time  does  not  vanish  for  all  these 
values  of  h.  This  function  has  a  smallest  value  or  a  minimum 
if  under  the  same  conditions  there  exists  the  difference 

[2]  /(*0+*)-/(*o)sO, 

which  does  not  vanish  for  all  values  of  h  between  —  8  and  +  B. 

A  function  may  have  several  such  maxima  and  minima  which 
may  be  different  from  one  another ;  it  may  have  minima  which  are 
greater  than  maxima.  (See 
the  accompanying  figure.) 
When  the  existence  of  com 
plete  derivatives  in  the 
entire  interval  under  con 
sideration  is  presupposed, 
the  maxima  and  minima 
which  may  be  derived  are  called  ordinary,  but  when  we  have  to  do 
with  functions  whose  derivatives  exist  only  on  definite  points  or  with 
functions  which  have  one-sided  derivatives  and  the  like,  the  maxima 
and  minima  may  be  called  extraordinary.  The  discussion  will  be 

*  See  Genocchi-Peano,  Calcolo  differenziate  e  principii  di  calcolo  integrate  (§  131). 

'1 


2  THEORY  OF  MAXIMA  AND  MINIMA 

restricted  at  first  to  ordinary  maxima  and  minima.  A  maximum  of 
f(x)  is  called  proper  by  Stolz  (G-rundzuge  der  Differential-  und  Inte- 
gralrechnung,  Vol.  I,  p.  199)  if  in  the  formula  [1]  only  the  sign  < 
stands ;  while  we  have  a  proper  minimum  if  there  stands  only  the 
sign  >  in  [2].  The  maximum  and  minimum  are  improper  if  in  formu 
las  [1]  and  [2]  the  sign  =  also  appears,  however  small  8  may  be  taken. 

/          1\2  1 

For  example,  y  =  (  x  sin  -  I  has  the  value  +  0  when  x  =  —  for 
\          a?/  nir 

consecutive  integral  values  of  n,  however  large,  and  that  is  for 
intervals  as  small  as  we  wish.  Stolz  and  others  *  use  the  notation 
extreme  or  extreme  value  of  a  function  to  denote  indifferently 
either  the  maximum  or  minimum  of  the  function. 

The  maximum  and  minimum  of  a  function  defined  as  above 
are  often  denoted  as  absolute  ~\  maximum  and  minimum,  since 
they  depend  upon  the  collectivity  of  the  values  of  f(x).  Opposed 
to  them  appear  the  relative  maximum  or  minimum,  which  enter 
if  the  independent  variable  x  is  subjected  to  a  restriction  so  that 
h  in  the  formulas  [1]  and  [2]  can  take  only  restricted  (and  not 
arbitrary)  positive  and  negative  values. 

2.  If  the  function  f(x)  has  for  x  =  XQ  a  positive  derivative 
ff(xQ),  the  function  is  becoming  larger  on  this  position  with  in 
creasing  a?,  and  its  values  are  respectively  smaller  or  greater  than 
those  o£/(a;0)  according  as  x  is  smaller  or  greater  than  XQ.  It  is 
assumed  that  x  lies  sufficiently  near  XQ. 

In  this  case  the  function  f(x)  has  for  x  =  XQ  neither  a  maximum 
nor  a  minimum.  Similar  (mutatis  mutandis)  conclusions  are  drawn 
if  /'(XQ)  is  negative. 

It  follows  that  if  the  function  f(x)  has  for  x  =  x0  a  finite  de 
rivative  that  is  different  from  zero,  then  on  this  position  the  function 
has  neither  a  maximum  nor  a  minimum. 

If  then  we  exclude  from  the  values  of  x  those  to  which  a  defi 
nite  derivative  (different  from  zero)  corresponds,  there  remain  either 

*  Extremer  Werth  was  used  by  R.  Baltzer,  Elem.  d.  Math.,  Bd.  I,  Aufl.  5,  S.  217; 
Extremum  by  P.  du  Bois-Reymond,  Math.  Ann.,  Vol.  XV,  p.  564. 

t  The  authors  just  cited,  as  also  Peano,  understand  by  the  absolute  maximum  and 
minimum  of  a  function  in  a  given  interval  the  upper  and  lower  limits  of  the  function  in 
this  interval,  if  such  limits  are  reached .  See  also  A.  Mayer,  Leipz.  Ber.  (1899) ,  p.  122,  and 
Lipschitz,  Analysis,  Vol.  II,  p.  306,  and  in  particular  Voss,  Encyklopddie  der  Math.  Wiss., 
Bd.  II,  Theil  I,  Heft  I,  S.  80,  who  remarks  upon  the  weak  terminology  of  the  subject. 


FUNCTIONS  OF  ONE  VARIABLE          3 

those  positions  on  which  the  function  has  no  derivative  (finite  or 
infinite)  or  those  places  on  which  it  has  a  vanishing  derivative. 

These  positions  must  be  further  examined  if  we  wish  to  make 
ourselves  sure  of  the  existence  or  nonexistence  of  a  maximum  or 
minimum.  No  rule  can  be  given  for  the  cases  where  derivatives 
do  not  exist. 

If  we  assume  that  the  derivative  is  zero,  the  following  criteria 
may  be  used  :  If  f(x)  has  the  derivative  /'(/-•)  in  the  interval 
(x0  —  h  •  •  •  XQ  H-  h),  we  have,  in  virtue  of  the  Taylor  formula,* 


where  xl  lies  between  #0  and  x.  If  f'(x)  becomes  zero  on  the  posi 
tion  x  =  XQ  in  such  a  way  that  it  is  positive  for  x  <  XQ  and  negative 
for  x  >  xQ,  then  (x  —  x^f'(x^  is  always  negative,  however  x(=£  XQ) 
be  taken,  and  consequently  it  follows  that  /(a?)  </(#0)  for  all  values 
of  x  within  the  interval  XQ  —  li  to  XQ  -f  h.  The  function  will  there 
fore  be  in  this  case  a  proper  maximum  for  x  =  XQ.  If,  however, 
ff(x)  is  negative  for  X<XQ  and  positive  for  x>xQt  the  product 
(x  —  #())/'(#!)  is  always  positive,  and  the  function  will  therefore 
be  a  proper  minimum  for  x  =  XQ  within  the  interval  in  question. 

If  f'(x)  is  zero,  say,  for  values  of  x  within  the  interval  XQ  •  •  •  xQ+h 
or  within  the  interval  XQ  —  h  •  •  •  .  x0,  we  have  cases  of  improper  ex 
tremes  (maxima  or  minima).  But  if  f'(x)  retains  a  constant  sign 
in  the  neighborhood  of  x  =  XQ,  then  (x  —  xQ)f'(x1)  changes  its 
sign  according  as  X>XQ  or  X<XQ,  and  the  function  has  neither  a 
maximum  nor  a  minimum. 

It  is  thus  seen  that  the  function  f(x)  has  on  the  position  x  =  XQ 
a  maximum  or  a  minimum  according  to  the  manner  in  which 
its  derivative  vanishes  for  x  =  XQ  ;  that  is,  according  as  we  pass 
from  positive  to  negative  values  or  from  negative  to  positive  values 
with  increasing  x.  It  has  neither  a  maximum  nor  a  minimum 
if  the  derivative  does  not  change  its  sign.} 

*  See  Pierpont,  The  Theory  of  Functions  of  Real  Variables,  Vol.  I,  p.  248. 
t  Leibniz,  Vol.  V,  pp.  220-226,  is  the  first  who  made  a  distinction  between  maximum 
and  minimum.    See  Maclaurin,  A  Treatise  of  Fluxions  (1742),  Vol.  I,  chap.  Lx,  and 

Vol.  II,  p.  695;  and  also  Cauchy,  Calc.  differ.,  p.  63.  With  Leibniz,  when  ^  —  0,  y  is 

ax 

a  maximum  if  the  curve  is  concave  towards  the  z-axis,  a  minimum  if  the  curve  is 
concave  away  from  the  x-axis. 


4  THEORY  OF  MAXIMA  AND  MINIMA 

3.  Instead  of  considering  the  sign  of  the  derivative  in  the 
neighborhood  of  xQt  if  we  consider  the  sign  of  the  second  deriva 
tive  for  x  —  XQ  (when  this  second  derivative  is  different  from  zero), 
we  have  the  rule  : 

The  function  f(x)  has  on  the  position  X  =  XQ  for  which  fr(xQ)  =  0 
a  maximum  or  a  minimum  according  as  f"(xQ)  is  negative  or 
positive.  Infinite  values  are  always  included  unless  it  is  stated 
to  the  contrary. 

In  fact,  if  f"(x())<Q,  then  f'(x)  is  a  decreasing  function,  and 
since  it  is  zero  for  x  =  XQ,  it  goes  from  positive  to  negative  values  ; 
the  inverse  is  the  case  if/"(x0)>0. 

This  rule  leaves  one  in  the  lurch  if  /"(aj0)=  0. 

If  in  general  it  is  assumed  that 

/'K)=o,  /"W=o,   ....  /(-«(«„)  =o, 

it  follows  from  Taylor's  formula  that 


where  xl  is  situated  between  x0  and  x. 

As  here  f(n\x)  is  assumed  to  be  a  continuous  function,  it  retains 
a  constant  sign  in  the  neighborhood  of  XQ.  If  n  is  odd,  the  factor 
(x  —  xQ)n  changes  sign  according  as  x  >  XQ  or  x  <  XQ.  Consequently 
f(x)—f(x0)  also  changes  its  sign  and/(a;0)  is  neither  a  maximum 
nor  a  minimum.  If  n  is  even,  the  factor  (x  —  XO)H  is  positive  and 
f(x)—f(x0)  has  always  the  sign  of  f(n\x^.  It  follows  that  f(xQ) 
is  a  maximum  or  minimum  according  as  /(w)(^j)  is  negative  or 
positive.  We  therefore  have  the  theorem  :  * 

If  for  x  =  XQ  the  first  and  some  of  the  following  derivatives 
vanish,  then  f(xQ)  is  or  is  not  an  extreme  according  as  the  first 
nonvanishing  derivative  is  of  even  or  odd  order.  If  it  is  of  even 
order,  there  is  a  maximum  or  a  minimum  according  as  the 
derivative  in  question  is  negative  or  positive. 

*  See  Maclaurin,  A  Treatise  of  Fluxions,  Vol.  I,  p.  226  ;  Vol.  II,  p.  695  ;  and  also 
Lagrange,  (Euvres,  Vol.  I  (1759),  p.  4.  It  was  Maclaurin  who  first  gave  a  correct 
method  of  distinguishing  maxima  from  minima. 


FUNCTIONS  OF  ONE  VAEIABLE         5 

4.  The  following  may  be  regarded  as  a  resume  of  what  has 
been  given  above  :  The  function  f(x)  is  supposed  to  be  uniquely 
denned  for  all  values  of  x  within  an  interval  (a,  b),  and  XQ  is  a 
point  of  this  interval.  The  function  f(x)  is  a  proper  maximum 
or  a  proper  minimum  for  x  =  XQ  if  we  are  able  to  find  a  positive 
number  8  sufficiently  small  that  the  difference  f(zQ  +  h)—f(xQ) 
retains  a  constant  sign  when  h  varies  from  —  8  to  +  &  If  this  dif 
ference  is  positive,  the  function  f(x)  is  smaller  for  x  =  XQ  than  it  is 
for  the  values  of  x  neighboring  XQ  ;  it  is  then  a  proper  minimum. 
On  the  contrary,  when  the  difference  /(«t'0+&)—/(#0)  is  negative, 
the  function  is  a  proper  maximum  for  x  =  XQ.  If,  furthermore,  the 
sign  =  enters  in  the  cases  just  mentioned,  however  small  8  be 
taken,  we  have  an  improper  minimum  or  maximum. 

When  the  function  f(x)  admits  a  derivative  for  the  value  #0  of 
the  variable,  this  derivative  must  be  zero.  In  fact,  the  two  quotients 

/(s0+a)-/(s0)  f(xQ-h)-f(x0) 

h  -h 

which  have  here  by  hypothesis  the  same  limit  /'(#<>)»  wnen  ^  tends 
towards  zero,  are  of  different  sign  ;  it  is  necessary  then  that  their 
common  limit  ff(xQ)  be  zero. 

Inversely,  let  XQ  be  a  root  of  the  equation  f'(x)=Q,  situated 
between  a  and  b,  and  taking  the  general  case  suppose  that  the 
first  derivative  which  is  not  zero  for  x  =  XQ  is  the  derivative  of 
the  nth  order  and  that  this  derivative  is  continuous  in  the  neigh 
borhood  of  the  value  XQ.  The  general  formula  of  Taylor  gives 
here,  limiting  it  to  the  term  of  the  ?ith  degree, 

/(*„+  *)  -/W  =    /<n)  K+  Oh)       (o  <  e  <  i) 


where  e  is  a  quantity  that  is  indefinitely  small  with  h.  Let  8  be  a 
positive  number  such  that  as  x  varies  from  x0  —  8  to  XQ  +  S  the 
absolute  value  of  e  is  smaller  than/(n)(#0),  so  that  /(#0  -{-&)—  /(#0) 

hn 

has  the  same  sign  as  —  /^(zA    If  n  is  odd,  we  note  that  this  dif- 
n  ! 

ference  changes  sign  with  h  ;  there  is  then  neither  a  maximum 


6 


THEOEY  OF  MAXIMA  AND  MINIMA 


nor  a  minimum  for  x  =  #0.  If  n  is  even,/(a30  +  h)  —  f(xQ)  has  the 
same  sign  as  /(H)  (#0)  whether  &  be  positive  or  negative ;  the  func 
tion  is  a  minimum  if/(w)(^0)  is  positive,  and  a  maximum  if/(?l)(#0) 
is  negative.  It  follows  that  for  the  function  to  be  a  maximum  or 
minimum  for  x  =  XQ  it  is  necessary  and  sufficient  that  the  first 
derivative  which  vanishes  for  x  =  #0  be  of  even  order.* 

In  geometric  language  the  preceding  conditions  denote  that  the 
tangent  to  the  curve  y=f(x)  at  the  point  J^  is  parallel  to  OX 
and  is  not  an  inflectional  tangent  (see  Figs.  2-5). 


o 


-X 


FIG.  2 


FIG.  3 


FIG.  4 


X0 

FIG.  5 


-X 


II.    EXTRAORDINARY  MAXIMA  AND  MINIMA 

A.   FUNCTIONS  WHICH  HAVE  DERIVATIVES  ONLY  ON 
DEFINITE  POSITIONS 

5.  Let  the  function  y=f(x)  be  uniquely  defined  for  all  values 
of  x  between  XQ  —  8  and  #0  +  8  and  suppose  that  it  is  continuous 
for  x  =  XQ.  If  the  expressions 


h 


—  h 


h> 


*  See  Goursat,  Cours  D'  Analyse,  Vol.  I,  pp.  108  et  seq.  I  shall  refer  hereafter  to 
this  work  by  the  name  of  the  author,  and  by  Peano  and  Stolz  I  shall  designate  the 
works,  cited  above,  of  these  two  mathematicians. 


FUNCTIONS  OF  ONE  VARIABLE  7 

have  limiting  values  when  lim  h  =  +  0,  each  of  these  expressions 
is  called  a  one-sided  differential  quotient,*  the  first  the  right-hand, 
and  the  second  the  left-hand,  differential  quotient  of  f(x)  with 
regard  to  x  for  the  value  x  =  x0. 

If  the  two  one-sided  differential  quotients  of  /(#)  are  equal  to 
each  other  for  x  =  xQt  their  common  value  is  called  the  complete 
differential  quotient  of  f(x)  with  respect  to  x  f  or  x  =  XQ. 

If  next  it  is  assumed  that  the  one-  valued  function/"^)  is  con 
tinuous  for  all  values  of  the  interval  (a,  b)  and  has  at  least  one 
sided  differential  quotients,  the  differential  calculus  offers  a  method 
for  the  determination  of  the  maxima  and  minima.  For  if  f(xQ)  is 
such  an  extreme  of  f(x),  the  quotient 


h 

must  necessarily  either  vanish  or  change  sign  with  k. 

It  may  therefore  be  concluded,  as  in  §  2,  that  the  complete 
differential  quotient  ff(x)  must  be  zero,  and  that  if  the  right- 
hand  and  left-hand  differential  quotients  of  f(x)  are  different  at 
the  point  x  =  XQ,  they  cannot  have  the  same  sign.  These  require 
ments  are  under  the  existing  conditions  necessary  that  f(xQ)  be 
an  extreme  of  f(x)  ;  however,  as  it  will  be  seen  in  the  following, 
they  are  not  always  sufficient. 

6.  Criteria  as  to  whether  a  root  x  =  XQ  of  the  equation  ff(x)=  0 
offers  an  extreme  of  the  function  f(x}.\ 

THEOREM  I.  If  /'(#)  vanishes  for  x  =  xQt  and  if  a  positive 
quantity  8  may  be  so  chosen  that  f(x)  has  complete  differential 
quotients  in  the  interval  (XQ  —  S  •  -  •  XQ+  8),  and  if  f'(x)  changes 
sign  neither  in  the  interval  (#0—  8  •  •  •  XQ)  nor  in  the  interval 
(x0  •  -  -  XQ  +  8)  and  also  does  not  remain  invariably  zero  in  either 
of  these  intervals,  then  f(xQ)  is  or  is  not  a  proper  extreme  of 
f(x)  according  as  the  sign  of  f(x)  hi  the  first  interval  is  different 
from  or  the  same  as  it  is  in  the  second  interval;  and  further 
more,  in  the  first  case  f(xQ)  is  a  maximum  or  a  minimum,  of 

*  See  P.  du  Bois-Reymond,  Math.  Ann.,  Vol.  XVI,  p.  120  ;  see  also  Pierpont,  The 
Theory  of  Functions  of  Real  Variables,  Vol.  I,  p.  223. 

t  See  Cauchy,  Calc.  differ.,  Lesson  7,  and  see  in  particular  Stolz,  pp.  201-210. 


8  THEOEY  OF  MAXIMA  AND  MINIMA 

f(x)  according  as  f'(x)  on  its  passage  through  zero,  as  x  with 
increasing  values  passes  through  xQt  changes  from  the  sign  -f-  to 
the  sign  —  or  from  the  sign  —  to  the  sign  +. 

This  theorem  is  stated  at  the  end  of  §  3  and  there  proved  ;  and 
as  also  indicated  in  that  section,  the  inconvenience  arising  due 
to  the  consideration  of  the  sign  f'(x)  may  be  obviated  if  the  func 
tion  f(x)  has  a  complete  second  differential  quotient  for  x  =  XQ. 
This  leads  to  the  following  theorem  : 

THEOREM  II.  If  under  the  conditions  assumed  in  Theorem  I 
the  function  f(x)  has  for  x  —  XQ  a  complete  second  differential 
coefficient  /"(#0)  which  is  not  zero,  then  f(xQ)  is  a  proper  ex 
treme  of  /(#),  being  a  maximum  or  minimum  according  as  frt(xQ) 
is  negative  or  positive. 

Due  to  the  definition  of  a  complete  second  derivative  it  follows 

that 


where  RQi)  is  a  quantity  that  becomes  indefinitely  small  with  h. 
If  here  the  existence  of  a  second  derivative  of  f(x)  is  assumed 
only  for  x  —  xQ9  then,  since  f'(xQ)  =  0,  there  corresponds  to  every 
positive  quantity  e  another  quantity  8  such  that,  if  —  8  <  h  <  8, 
we  have  ,  -,, 


If,  say,  f"(xQ)  is  positive,  it  follows  at  once  that  there  must  be 
a  positive  quantity  8  such  that  for  —  8  <  h  <  8  we  have 

[3]  £/'(««+  *)>0. 

Hence  f'(xQ+h)  must  be  negative  when  h  is  negative  and  posi 
tive  when  h  is  positive,  so  that  on  passing  through  zero  (i.e.,  when 
x  —  a;0),  ff(xQ+  h)  passes  with  increasing  x  from  a  negative  value 
to  a  positive  value.  Accordingly,  in  virtue  of  Theorem  I,  f(xQ)  is 
a  proper  minimum. 

The  above  theorem  becomes  the  one  given  in  §  3  if  it  is 
assumed  that  there  is  an  interval  including  the  value  x  =  #0  such 
that  for  all  points  within  it  a  second  differential  quotient  of  f(x) 


FUNCTIONS  OF  ONE  VARIABLE         9 

exists,  and  if  it  is  further  assumed  that  f"(x)  is  continuous  at 
least  at  the  point  x  =  XQ. 

THEOREM  III.  If,  furthermore,  f"(x0)  =  0  and  /'"(#o)  ^  °>  then 
f(xQ)  is  not  an  extreme  of  /(#). 

For  since  here 


it  is  seen  that  as  ff(x)  passes  through  the  value  /'(#0)  =  0,  it  does 
not  change  sign,  and  consequently  /(#0)  is  not  an  extreme  of  f(x). 

The  two  preceding  theorems  are  special  cases  of  the  two 
following  : 

THEOREM  IV.    If  for  the  value  x  =  x0  we  have 


then  /(JJQ)  is  a  proper  extreme  of  f(x),  being  a  minimum  or  maxi 
mum  according  a,sfW(xQ)  is  positive  or  negative. 

For,  owing  to  the  supposed  existence  of  the  first  2  k  differential 
quotients,  there  is  an  interval  XQ  —  8  •  -  •  XQ  +  8  throughout  which 
the  differential  quotients  /'(#),/"(«),  •  •  •,/(2A'~1)(^)  not  only  exist 
but  are  also  continuous  functions  of  x.  Accordingly,  in  virtue  of 
Taylor's  formula,  we  have 

>2k-l 

[4]    /(**  +  ft)  -/(* 


(2     _  l 

which  formula  is  true  for  all  values  of  h  such  that  —  8  <  h  <  8. 
Owing  to  the  existence  oifW(xQ),  as  in  the  case  of  formula  [3] 
above,  it  is  seen  that  for  values  of  h  such  that  —  8  <  h  <  8  we  have 

l/<2t-D(a.  +  h)>0     or   <0 
/?/ 

according  as/(2t)(#0)  is  positive  or  negative. 

If,  then,  /(2Xr)(#0)  is,  for  example,  positive,  it  is  clear  that 
/(2*)(a?0+  /i)  is  negative  for  values  of  h  in  the  interval  —  8  ...  0 
and  positive  for  values  of  h  in  the  interval  0  •  •  •  <X 

It  follows  from  [4]  that  the  difference  /(#0+  ^)—/(#o)  f°r  all 
values  of  A-  within  the  interval  —  8  ...  -h  8  (excepting  A=  0)  is  in 
variably  -f  or  —  according  as/(2Ar)(#0)  is  -h  or  —  ,  and  correspond 
ingly  we  have  respectively  a  proper  minimum  or  a  proper  maximum. 


10  THE011Y  OF  MAXIMA  AND  MINIMA 

If  it  is  further  supposed  that  f(2k)(x)  exists  for  all  values  of  x 
in  the  interval  XQ—  $  •  •  •  XQ  +  8  and  that  fW(x)  is  a  continuous 
function  at  least  at  x  =  XQ,  then,  as  in  §  3,  due  to  Taylor's 
expansion  we  have 

[5]          /K  +  A)-/ 


from  which  the  theorem  is  obvious. 

In  exactly  the  same  way  we  may  prove 

THEOREM  V.  If  for  x  =  XQ  the  2  k  first  differential  quotients  of 
f(x),  viz.,  /'(a0),  /"(a0),  •  •  .,  /0*>(a>0),  vanish,  and  if  /<**+1>(a0)  *=  0, 
then  f(xQ)  is  not  an  extreme  of  /(#). 

REMARK.  In  the  case  that  a;  =  x0  causes  every  differential  quotient  of 
f(x)  to  vanish,  we  cannot  determine  by  means  of  Theorems  II,  III,  and 

IV  whether  f(x)  is  an  extreme  or  not.    We  must  then  apply  Theorem  I. 

_  j^ 

For  example,  it  is  seen  that  x  =  0  is  a  minimum  of  f(x)  =  e   x*. 

THEOREM  Va.  If  the  given  function  f(x)  can  be  developed  in 
the  neighborhood  of  the  point  XQ  in  a  series  in  integral  positive 
powers  of  x  —  x0  =  h  so  that 

f(x)  =f(xQ  +  x  -  aj0)  =  cmhm  +  cm+1hm+*  +  •  •  .,  (cm  *  0) 

then  f(xQ)  is  not  an  extreme  of  f(x)  or  is  an  extreme  of  f(x) 
according  as  m  is  odd  or  even  ;  and  f(xQ)  is  a  maximum  or  mini 
mum  according  as  cm  is  negative  or  positive. 

For  here  f(x0)  =  0  =f"(x0)  .  .  .  .  =/<*-  D  (^0), 

while  /(w>  (a?0)  =  m  !  cw. 

This  theorem  may  be  proved  directly  by  means  of  the  property 
of  series.  For  under  the  given  assumptions  corresponding  to  every 
quantity  e  >  0,  we  may  choose  another  quantity  8  >  0,  such  that 

-  e  <  cm  +  1h  +  cm+Ji*  H  ----  <  e. 

If  m  is  a  positive  integer  and  cm  <  0,  say,  and  if  |  X  |  <  8, 
then  f(xQ  +  h)  -  f(x0)  <  k™  (cm  +  e)  ; 

and  as  e  may  be  taken  such  that  e  <  —  cm,  the  expression  on  the 
right-hand  side  is  always  negative,  so  that  there  is  a  maximum  of 
f(x)  at  x  =  xQ.  Similarly,  we  may  prove  the  remaining  part  of  Va. 


FUNCTIONS  OF  ONE  VARIABLE  11 

B.  FUNCTIONS  WHICH  HAVE  ONLY  ONE-SIDED  DIFFERENTIAL 
QUOTIENTS  OF  A  CERTAIN  ORDER  FOR  A  VALUE  x  =  x0* 

7.  THEOREM  VI.  If  the  continuous  function/^1)  has  for  x  =  x0 
one-sided  differential  quotients  of  the  first  order  and  of  opposite 
sign  (including  -f-  co  and  —  oc),  then  f(x0)  is  a  proper  extreme, 
being  a  maximum  or  minimum  according  as  the  right-hand 
differential  quotient  of  f(x)  is  negative  or  positive. 

For  if,  say,  the  left-hand  differential  quotient  is  positive,  the 
right-hand  one  being  negative,  then  there  exists  a  positive  quantity 
B  such  that  according  as  —  S  <  h  <  0  or  0  <  h  <  S,  we  have 


h 

It  follows  that  /(a;0+  h)  —  f(xQ)  <  0  for  all  values  of  h  that  are 
situated  within  the  interval  —  8  ...  +  S.  Hence  f(xQ)  is  a  proper 
maximum. 

THEOREM  VII.  If  for  x  =  XQ  the  2  k  first  differential  quotients 
of  /(<£),  viz.,  /'(a;),  /"(a,-),  .  .  •  ,  /<**>(,»),  vanish,  and  if  /<**>(,»)  has  for 
X  =  XQ  one-sided  differential  quotients  of  contrary  sign  (+00  and 
—oo  included),  then  the  value  /(#0)  forms  a  proper  extreme  of 
f(x),  being  a  maximum  or  a  minimum  according  as  the  right- 
hand  differential  quotient  is  negative  or  positive. 

If,  for  example,  the  left-hand  differential  quotient  is  positive, 
while  the  right-hand  is  negative,  that  is, 


—  h 

we  note,  since  fW(xQ)=  0,  that/^^-h  h)  <  0  for  all  values  of  h 
within  the  interval  —  8  •  •  •  +  8  (the  value  h  =  0  excepted).  Hence 
from  formula  [5],  viz., 


it  is  seen  at  once  that  f(xQ)  is  a  proper  maximum. 

THEOREM  VIII.    If    for    x  =  XQ   the    2k  —  I    first    differential 
quotients  vanish,  viz.,  f'(xQ)=f"(x()}=  .  .  .  =/(2*-D  =  0,  and   if 

*  Stolz,  p.  206;  see  also  Pascal,  Exercici,  etc.,  pp.  215-222.   1895. 


12  THEORY  OF  MAXIMA  AND  MINIMA 


-!)(#)  has  for  x  =  x0  one-sided  differential  quotients  of  oppo 
site  sign  (+  oo  and  —  co  included),  then  /(#0)  does  not  form  an 
extreme  of  /(#).  If,  however,  these  differential  quotients  are  both 
positive  or  both  negative,  then  /(#0)  is  a  proper  minimum  or 
maximum  of  f(x).  This  theorem  follows  from  [4]  in  the  same 
manner  as  the  preceding  one  did  from  [5]. 

Example.  If  f(x)  =  xv-  (x  ^  0)  and  /(>•)  =  (—x)v(x<  0),  show  by  means 
of  Theorem  VI  that  there  is  a  proper  minimum  at  x  =  0  if  /x  lies  be 
tween  0  and  +  1.  Verify  the  same  result  when  /x  lies  between  2  A: 
and  2  k  +  1  by  making  use  of  Theorem  VII  ;  and  by  using  Theorem  IV 
show  that  f(x)  is  a  proper  minimum  when  /x  is  situated  between  2  k  —  1 
and  2k. 


C.    UPPER   AND    LOWER    LIMITS    OF   A   ONE- VALUED    FUNCTION 

WHICH  is  CONTINUOUS  FOR  VALUES  OF  THE  ARGUMENT  WITHIN 

A  DEFINITE  INTERVAL 

8.  If  the  function /(x)  is  continuous  and  uniquely  defined  in 
the  definite  interval  (a,  b),  there  exist  the  greatest  and  the  least 
value*  in  the  interval  in  question,  which  are  known  as  the  upper 
and  lower  limits  of  the  function  in  this  interval ;  and,  further,  the 
function  reaches  these  limits ;  that  is,  if  these  limits  are  denoted 
by  g  and  Jc,  then  there  is  at  least  one  value  c  of  x  within  the 
interval  a  •  -  •  b  for  which  the  function  is  equal  to  gt  and  at  least 
one  value  d  within  the  same  interval  for  which  the  same  function 
is  equal  to  Jc. 

But  if  the  interval  within  which  x  varies  is  indefinitely  large, 
(a,  co)  or  (—  oo,  b)  or  (—  oo,  +  co),  the  function  need  not  have  a 
maximum  or  a  minimum,  and  also  it  need  not  have  an  upper  or  a 
lower  limit.  This  is  illustrated  in  the  following  examples.!  (See 
also  §  96.) 

*  Proofs  of  this  and  the  following  statements  are  found,  for  example,  in  Harkness 
and  Morley,  Intr.  to  Analytic  Functions,  §§46,  50;  E.  B.  Wilson,  Advanced  Cal 
culus,  §§  19-25.  See  Peano,  Theorem  IV,  §  21,  and  also  Dini,  Fundamenti  per  la 
teorica  delle  funzioni  di  varidbili  reali  (German  translation  by  Liiroth  and  Schepp, 
§§  36,  47).  These  proofs  are  founded  upon  Weierstrass's  lectures,  which,  in  turn, 
are  founded  upon  the  work  of  Bolzano,  Abh.  d.  Bohmischen  Gesellsch.  der  Wiss., 
Vol.  V,  p.  17. 

t  Peano,  §  132. 


FUNCTIONS  OF  ONE  VARIABLE         13 

Example  1.  Divide  a  number  into  two  parts  so  that  their  product  is  a 
maximum.  (Cf.  Ex.  6  at  end  of  §  10.) 

Let  a  be  the  given  number,  x  and  a  —  x  the  two  summands,  and 
y  —  (a  —  x)  x  their  product.  If  we  consider  x  as  variable,  we  have  y'  =  a—2x, 
which  becomes  zero  for  x  =  ^  •  We  further  have  /'  =  —  2,  so  that  the 
function  y  has  a  maximum  for  x  =  ^  ;  that  is,  when  both  parts  are  equal, 
this  value  being  y  =  —  - 

Since,  however,  the  derivative  /  is  positive  for  x  <  -  and  negative  for 
x  >  ->  it  follows  that  the  function  increases  in  the  interval  f  —  x,  -J  and 

decreases  in  the  interval  (-  ,  +  x  j.    The  function  has  neither  an  upper  nor 
a  lower  limit. 

Example  2.    y  =  a?r.          (x  >  0) 

Through  differentiation  we  have  /  =  j^(l  +  loga:).  The  first  factor 
is  never  zero  and  is  always  positive.  The  second  factor  becomes  zero 

when  log  x  =  —  1  or  x  =  -  •    The  derivative  passes  therefore  from  negative 
(for  x  <  -\  to  positive  values  (for  x  >  —  V    The  function  has  a  minimum 

for  x  =  -  =  0.36788  •  •  •,  which  is  y  =  0.676411  •  •  •.    This  is  also  the  lower 


e 


limit  which  the  function  takes  in  the  interval  (0,  x).    The  function  does 
not  have  either  a  maximum  or  an  upper  limit. 

Example  3.     y  =  art,     y'  =  §  x~  i 

The  derivative  is  zero  for  no  finite  value  of  x,  but  is  infinite  for  x  =  0. 
For  this  value  y  becomes  zero,  and  the  function  will  have  at  this  point  both 
a  minimum  and  a  lower  limit  with  respect  to  the  interval  (—  x,  +  oc);  for 
all  the  values  of  x  cause  the  function  to  be  greater  than  zero.  The  function 
has  neither  a  maximum  nor  an  upper  limit. 

9.  If  we  add  to  the  postulates  already  made  in  the  previous 
article  regarding  f(x)  that  it  must  have  a  complete  differential 
quotient  for  all  values  of  x  between  a  and  b,  then  /'(#)  vanishes 
for  every  value  of  x  between  a  and  b  to  which  one  of  the  values  g 
or  k  of  the  function  belongs.  If,  however,  /(a)  =  g,  say,  then  possibly 
/(a)  is  only  a  one-sided  maximum  of  /(£),  and  consequently  /'(a) 
is  not  necessarily  zero.  This  must  be  borne  in  mind  as  we  proceed 
with  the  problem  of  determining  the  numbers  g  and  X\  This  is 


14  THEORY  OF  MAXIMA  AND  MINIMA 

explained  by  the  simple  example  (see  Liouville's  Journal,  First 
Series,  Vol.  VII,  p.  163): 

In  the  plane  of  a  circle  which  is  described  about  the  point  0  as 
center  with  radius  r,  let  there  be  given  an  arbitrary  point  A  which 
is  different  from  0.  Determine  the  upper  and  lower  limits  of  the 
distances  of  the  point  A  from  a  point  M  of  the  circumference. 

Let  the  positive  X-axis  be  taken  passing  through  0,  and  standing 
perpendicular  to  it  through  0  is  erected  the  Y-axis.  The  equation 
of  the  circle  is  then  *2  4-  y*  —  r2 ;  while 

AM2  =  (a  -x)*+y*=r*+  a2  -  2  ax.  (a) 

As  M  passes  over  all  points  of  the  circumference,  x  takes  the  values 
in  the  interval  —  r  •  -  •  +  r.  The  linear  function  (a)  decreases  with 
increasing  values  of  «/:,  its 
differential  quotient  being  a 
negative  constant  equal  to 

—  2  a.    Consequently  those 
values   of   x   to  which  the 
upper  and  lower  limits   of 
AM2   belong,    fall    on    the 
end-points    of    the   interval 

x  IG.  O 

—  r  •  •  •  H-  r.    It  is  seen  that 

—  r  corresponds  to  the  upper  limit  and  +  r  to  the  lower  limit,  giving 
us  as  upper  and  lower  limits  respectively   a  +  r  |  and  |  a  —  r  \ . 

10.  Suppose  next  that  the  function  f(x)  is  discontinuous  at  least 
on  an  end-point  of  the  arbitrary  interval  (a,  5);  for  example,  sup 
pose  that  the  function  is  not  denned  at  such  a  point.  If  this  is 
the  case  only  for  the  lim  x  =  at  then  in  the  derivation  of  the 
upper  and  the  lower  limit  we  must  consider  in  particular  the  value 
of  f(x)  for  the  lim  x  =  a  +  0.  The  following  examples  will  make 
clear  the  method  of  procedure  (see  Stolz,  p.  210). 

Example  1.    Consider  the  function  ?/  = for  values  of  x  such  that 

log* 
0  <  x  <  1.    It  is  seen  that  y  is  negative  and  decreases  with  increasing  values 

of*.    For  when  lim  *=  +  (),        then     limy=-0; 

and  when  lim  *  =  1  —  0,     then     lim  y  —  —  QC. 

Thus  the  upper  limit  of  y  in  this  interval  is  zero,  while  the  lower  limit  is 

—  x>,  although  neither  is  reached. 


FUNCTIONS  OF  ONE  VAKIABLE  15 

Example  2.  y  =  (1  -  x)  sin  -.         (0  <  x  ^  1) 

For  these  values  of  .r  we  have  always  |#j<l-  o 

If  we  consider  onlv  values  of  x  such  as  x  —  -  —  -  —  -  (where  n  is  an 

,  •  4  n  +  1 

integer),  we  have 


Hence  when  n  =  +  cc,  the  upper  limit  of  y  is  +  1. 

Bv  writing  x  =  —  -  —  ,  it  is  seen  that  the  lower  limit  is  —  1.    Neither 

4n  —  1 
the  upper  nor  the  lower  limit  of  y  is  reached,  although  in  either  case 

they  are  finite. 


PROBLEMS 
1.  Determine  the  maxima  and  minima  and  the  upper  and  lower  limits  of 


(b)  y  =  a  cosx  +  b  sin  x.  ex  +  1 

(c)  y  =  a  +  x*.    (Pierpont,  p.  320.)  -1 

(g)  y  =  x2-  —  ex*. 

(d)  y  =  l  —  x*.  (Maclaurin,  Vol.  IT,  l 

p.  720.)  (h)  ?  =  «"*. 

(e)  a-2sin_-     (The  function   has  a  -~i    ,,_, 

x  (i)  y  =  xe    «*.  (There  is  no  ex- 

discontinuous  derivative  for  treme  on  the  position 

x  =  0.)  x  =  0.) 

2.  Show  that  the  function  //<*>  =  *  Sin  f         ("  *  °} 

l/(0)  -  0 
has    an    infinite    number    of    maxima    and    minima   within    the    interval 


_          _ 

3.  When  is  mp  +  nq  a  minimum,  where  p  =  V<?2  +  if,  q  =  V£2  +  (h  —  y)2  ? 
(Leibniz,  1682.) 

4.  "In  venire  cvlindrum  maximi  ambitus  in  data  sphaera."    (Fermat, 
(Eurres,  Vol.  I,  p.  167.    1642.) 

5.  Find  the  area  of  the  greatest  parabola  which  may  be  cut  from  a 
given  cone. 

6.  x  (x  —  a)  has   its   greatest  value  when   x  =  -  •    (Euclid,  Book  VI, 

Prop.  27.)    Cantor  (Geschichte  der  Math.,  Vol.  I,  p.  228)  says  that  this  is  the 
first  example  of  a  maximum  in  the  history  of  mathematics. 


16  THEORY  OF  MAXIMA  AKD  MINIMA 

7.  On  a  given  line  AB  are  two  fixed  points  Pl  and  P2.    Determine  a 

AP  •  PS 
third  point  so  that  —^—^ — — — -  is  a  minimum.    (Pappus,  Book  VII,  Prop.  61, 

*y  *  ^  2 

and  Fermat,  CEuvres,  Vol.  I,  p.  140.) 

8.  Of  all  sections  which  pass  through  the  vertex  of  a  cone,  determine 
the  one  of  greatest  area.    (Severus.) 

•  9.  The  number  a  is  to  be  divided  into  two  parts,  such  that  their  product 
multiplied  into  their  difference  shall  be  a  maximum.  (Tartaglia,  General 
Trattato,  Part  5,  fol.  88.) 

10.  A  ten-foot  pole  hangs  vertically  so  that  its  lower  end  is  four  feet 
from  the  floor.    Find  the  point  on  the  floor  from  which  the  pole  subtends 
the  greatest  angle.    (Regiomontanus.    1471.) 

11.  The    curve   y  =  x2  — has   no   minimum.     (Euler,    Differential- 

rechnung,  Vol.  Ill,  p.  744.)       g  X 

12.  Two  points  Pl  and  P2  not  on  the  straight  line  AB  are  given.    Find 
a  point  P  on  AB  such  that  PP±  +  PP^  is  a  minimum.   (Solved  by  Huygens 
possibly  about  1673.    See  Huygens,  Opera  Varia,  pp.  490  et  seq.    Note  the 
letters  of  De  Sluse.) 

13.  Derive  the  greatest  rectangle  that  can  be  described  in,  and  having 
one  of  its  sides,  upon  the  base  of  a  given  triangle.    (Simpson,  Elements  of 
Plane  Geometry  (1747),  pp.  106  et  seq.    In  this  work  are  also  found  numerous 
problems  that  have  to  do  with  areas,  volumes,  etc.) 


CHAPTER  II 

FUNCTIONS  OF  SEVERAL  VARIABLES 
I.    ORDINARY  MAXIMA  AND  MINIMA 

PRELIMINARY  EEMARKS 

• 

11.  We  say  that  the  function  u  =f(xv  #2,  •  •  •,  xn)  becomes  a 
maximum  or  minimum  on  the  position  (av  a2, .  • .,  an)  if  for  a 
sufficiently  small  region  about  (alt  «2,  •  •  -,  an)  we  have 

f(av  a2,  •  •  .,  an)^f(xv  x2,  .  .  -,  xn) 
or  f(av  aa,  •  •  -,  an)  ^  f(xv  #2,  •  -  .,  xn). 

These  extremes  are  proper  or  improper  according  as  the  sign  = 
does  not,  or  does  enter. 

As  in  §  1,  it  is  assumed  here  that  the  function  has  definite 
partial  derivatives  which  are  continuous  within  the  region  in  ques 
tion  with  regard  to  each  of  the  variables ;  and  the  extremes  which 
may  be  derived  we  shall  call  ordinary.  If  the  partial  derivatives 
do  not  have  such  derivatives,  the  extremes  may  be  called  extraor 
dinary.  Such  extremes  in  their  generality  we  shall  not  attempt  to 
consider.  Another  class  of  extraordinary  extremes  is  mentioned 
in  §  13,  and  is  later  treated  in  its  generality  for  the  case  of  func 
tions  of  two  variables  (§§  20  et  seq.). 

12.  Consider  the   function   of   one   variable   xlt  viz.,  f(xv   av 
•  •  -,  an).    If  the  function  u  of  the  preceding  article  is  a  maximum 
or  minimum  for  x^=  al9  •  •  •,  xn  =  an,  then  f(xlf  a1?  •  •  -,  an)  will  be 
a  maximum  or  minimum  for  x1  =  av   Hence  (see  §  2)  the  derivative 
f'x  (xv  a2, .  .  .,  an)  must  be  zero.    Similar  conclusions  may  be  made 
for  the  other  variables  in  u. 

It  follows  that  if  u  =f(xv  •  •  •,  xn)  has  an  extreme  on  the  position 
(av  a2,  •  •  -,  an),  the  first  partial  derivatives  of  u  must  be  zero. 

17 


18  THEORY  OF  MAXIMA  AND  MINIMA 

(See  Euler,  Gale.  diff.  (1755),  p.  645;  and  Lagrange,  Theorie  des 
Fonctions,  Vol.  II,  No.  51.) 

Write*  next  xl=a1+h1tf  •  •  -,  a>,t=  a,t  +  hnt,  and  put 


If  w  =/(#!,  •  •  •,  xn)  is  an  extreme  on  the  position  (av  -  .  .,  rcM), 
then  F(t)  is  an  extreme  on  the  position  t  =  0. 

Since  by  hypothesis  the  derivatives  of  u  are  continuous,  it 
follows  also  that  the  same  is  true  of  F(t}. 

We  consequently  may  write 


It  follows  from      2  that 


whatever  be  the  values  of  hv  h2,  •  •  .,  hn. 
We1  therefore  have 

AK,  •  •  •,  «„)=  0,  .  .  .,/£„(«!,  •  •  -,  an)=  0, 
as  was  just  seen. 
We  further  have 


If  ^  is  to  be  an  extreme  for  the  position  under  consideration,  then 
F(t)  must  be  an  extreme  for  t  =  0,  so  that  for  a  maximum  we  must 
have  (§  3)  j^'(O)  =g  0,  and  for  a  minimum  ^"(0)  i=  0,  whatever  be 
the  values  of  hv  hz,  •  -  .,  hn.  If  for  the  time  being  we  omit  the  sign 
=  from  the  two  expressions  just  written,  we  have  the  theorem  : 

In  order  that  the  function  u  be  an  extreme  at  the  position 
(ai>  '  '  •>  an)  for  ^hich  the  first  derivatives  vanish,  it  is  necessary 
that  the  following  homogeneous  function  of  the  second  degree  in 
hi,  -  •  -,hn)  viz., 


See  also  Peano,  §  134. 


FUNCTIONS  OF  SEVERAL  VARIABLES      19 

assume  only  positive  or  only  negative  values,  whatever  be  the 
values  of  hv  -  •  •  ,  hn,  except  when  these  quantities  are  all  simul 
taneously  zero. 

13.  We  distinguish  three  kinds  of  integral  functions  of   the 
second  degree,  or  as  they  are  usually  called,  quadratic  forms*  viz., 

I.  Definite  forms,  which  with  real  values  of  the  variables  have 
always  the  same  sign,  that  is,  only  positive  values  or  only  negative 
values,  and  are  only  zero  when  the  variables  are  all  zero. 

II.  Semi-definite  forms,]   which  always  have    the    same   sign, 
but  which  vanish  also  for  other  values  of  the  variables  that  are 
not  all  zero. 

III.  Indefinite  forms,  w^hich  with  real  values  of  the  variables 
can  become  both  positive  and  negative,  and  that  too  for  values  of 
the  variables  whose  absolute  values  do  not  exceed  an  arbitrary 
small  quantity. 

The  theorem  of  the  preceding  section  may  be  written  as  follows : 
If  for  x1  =  av  •  -  .,  xn=  an  the  first  partial  derivatives  of  the 
function  u=f(xv  •  •  .,  xn)  vanish,  and  if  in  the  Taylor  develop 
ment  t  for  f(xl  -h  h !,-•-,  xn  4-  hn)  the  term  which  is  a  homogeneous 
function  of  the  second  degree  in  hv  •  •  •,  hn  is  an  indefinite  form, 
then  u  on  the  position  (#!,•••,  «„)  has  neither  a  maximum  nor  a 
minimum  value.  If,  on  the  other  hand,  that  term  is  a  positive  defi 
nite  form,  then  u  is  a  minimum,  and  if  it  is  a  negative  definite 
form,  u  is  a  maximum. 

The  case  where  the  form  is  semi-definite  is  included  under  the 
extraordinary  extremes,  and  we  shall  consider  it  later  (§§  20  et  seq.). 

14.  Next  is  given  a  criterion  to  determine  whether  a  given  quad 
ratic  form  <t>(hlt  •  •  -,  hn)  is  a  positive  definite  quadratic  form. 

If  (/>  depends  only  upon  one  variable  hv  we  shall  have  <f>=Ahf, 
and  this  is  positive  when  and  only  when  A  is  positive. 
If  <f>  depends  upon  two  variables  h1  and  7*2,  we  shall  have 

<£  =  AJil  +  2  Bh^  +  Chi 

*  See  Gauss,  Disq.  Arithm.,  p.  271. 

t  So  called,  for  example,  by  Scheeffer,  Math.  Ann.,  Vol.  XXXV,  p.  555.  Gergonne, 
Gerg.  Ann.,  Vol.  XX  (1830),  p.  331,  called  attention  in  particular  to  this  case. 
J  This  development  is  found  in  full  in  §  50. 


20  THEOEY  OF  MAXIMA  AND  MINIMA 

If  here  </>  is  a  positive  definite  form,  it  follows  that  for  h2  =  0, 
A!  =£  0,  then  (f)  =  Ahf,  and  consequently  A  must  be  positive.  We 
may  also  write  $  in  the  form 


If   in   this   expression   we  give  to  \  and  h2  such  values  that 

Ahl  4-  Bh%  =  0,  it  is  seen  that  <j>  takes  the  form  </>  =  —  (AC  —  B2)  hj. 
We  must  therefore  have  A  C  —  B2  >  0. 

The  conditions  A  >  0  and  AC  —  B2>0  are  not  only  necessary, 
but  they  are  also  sufficient  that  <£  be  a  definite  quadratic  form. 
In  fact,  if  &2=£0,  we  have  (AC-JBP)h$>Q  and  (^1  +  ^2)2-°» 
and  consequently  the  sum  of  these  two  expressions,  and  also  </>,  is 
positive. 

If,  in  general,  (f>  depends  upon  several  variables  hv  A2,  7i3,  •  •  ., 
we  may  write 


where  A  is  a  constant,  B  a  form  of  the  first  degree  in  h2,  ks,  •  •  ., 
and  C  a  quadratic  form  in  h2,  A3,  •  •  •. 

If  ^2,  7^3,  •  •  •  are  all  zero,  but  /^  ^  0,  we  will  have  B  and  C  zero 
and  <£  =  Akf.  We  must  therefore  have  A  >  0,  if  (/>  is  to  be  a 
positive  definite  form. 

The  form  may  be  written 


where  AC  —  B2  is  a  quadratic  form  of  h2,  hs,  •  •  •.    The  quantity 
Ax  may  be  determined  so  that  AJi^  +  B=Q  with  the  result  that 


Hence   the  expression  AC  —  B2  must  be  positive  and  different 
from  zero. 

Next  write  AC  —  &—  ^(h^  hs,  •  •  •),  where  c^  is  a  quadratic 
form  in  7&2,  A3,  •  -  -  which  is  always  positive  and  different  from 
zero  except  when  all  the  variables  vanish. 


FUNCTIONS  OF  SEVERAL  VARIABLES  21 

It  follows  that  the  necessary  conditions  that  (/>  be  a  definite 
positive  form  are  (1)  that  A  be  greater  than  0  and  (2)  that 
AC—  IP  be  a  positive  definite  form  in  the  variables  7t2,  hs,  •  •  .. 

These  conditions  are  also  sufficient;  for  if  we  give  to  hl  an 
arbitrary  value  and  to  A2,  h3,  •  •  •  arbitrary  values  which  are  not 
all  zero,  then  of  the  two  summands  into  which  <f>  is  distributed, 
the  first  is  positive  or  zero,  while  the  second  is  positive.  It  follows 
that  <f>  is  positive.  On  the  other  hand,  if  we  give  to  &2,  &3,  •  •  • 
simultaneously  the  value  zero,  then  Ji^  must  be  different  from 
zero,  and  from  (f>=Ahf  it  is  seen  that  A  must  be  positive.  In 
this  way  the  determination  of  the  question  whether  a  quadratic 
form  is  definite  and  positive  is  reduced  to  the  determination  of 
the  same  question  in  the  case  of  another  quadratic  form  of  fewer 
variables.  If  then  the  process  is  continued,  we  come  to  the  forms 
in  one  or  two  variables  already  considered.  This  subject  is  further 
considered  in  §§  53  et  seq. 

To  determine  whether  a  quadratic  form  <£  is  definite  and  nega 
tive,  we  have  to  determine  whether  —  <£  is  definite  and  positive. 
(See  Peano,  §  137.) 

II.    RELATIVE  MAXIMA  AND  MINIMA 

15.  To  introduce  the  theory,  we  shall  consider  here  a  simple 
case  involving  only  three  variables.  Let  it  be  required  to  deter 
mine  the  extremes  of  the  function 

u  =  F(x,  y,  z), 

where  the  variables  xy  y,  z  are  restricted.    Suppose,  for  example, 
that  they  are  connected  by  the  equation 

f(x,  y,  z)  =  0. 

If  from  the  latter  equation  z  is  expressed  as  a  function  of  x 
and  y,  and  if  this  value  is  substituted  in  the  first  equation,  we 
shall  have  u  expressed  as  a  function  of  x  and  y.  The  values  #,  y 
which  make  u  a  maximum  or  minimum  cause  the  total  derivative 
du  to  vanish  for  all  values  dx  and  dy. 


22  THEORY  OF  MAXIMA  AND  MINIMA 

dF          dF          dF 

We  have  du  =  — -  dx  +  —  dy  4-  — -  dzt 

dx  dy  dz 

where  dz  denotes  the  differential  of  zs  which  is  denned  through 
the  equation  $f  y  $f 

dx  dy    '        dz 

If  this  last  equation  is  multiplied  by  the  indeterminate  quan 
tity  —  X  and  added  to  the  equation  du  =  0,  we  have 


If  in  this  equation  we  choose  X  so  that  the  coefficient  of  dz 
vanishes,  then  corresponding  to  the  maxima  and  minima  values 
of  u  the  coefficients  of  dx  and  dy  must  also  be  zero,  and  we  thus 
have  the  equations 

£-x£=o,  g^!=o,  g-x|=o. 

dx         dx  dy         dy  oz          dz 

It  is  evident  that  we  have  these  expressions  which  are  sym 
metric  with  regard  to  the  three  variables  if  we  form  the  three 
partial  derivatives  of  F  —  X/,  where  X  is  an  indeterminate  quan 
tity,  and  then  put  the  resulting  expressions  equal  to  zero. 

These  three  equations,  together  with  the  two  equations  /=  0 
and  u=F,  determine  the  unknown  quantities  X,  x,  y,  z,  u,  which 
correspond  to  the  values  of  u  for  which  there  exist  maxima  and 
minima  values. 

We  may  proceed  in  the  same  manner  with  an  arbitrary  number 
of  variables  and  equations  of  condition.  (See  Lagrange,  Theorie  des 
Fonctions,  p.  268.) 

PROBLEMS 

1.  Find  the  minimum  value  of  u,  where 

M  =  a:2  +  y2  +  z2+  -.., 

and  where  x,  y,  z,  •  •  •  are  connected  by  the  equation 
ax  +  by  +  cz  +  •  •  •  =  k. 

2.  If  xl  +  £2  4-  •  •  •  +  xn  =  a,  show  that 

x?  +  x.}+  ->  +k£ 
is  a  minimum  when  x^  =  x<,  =  •  •  •  —  xn.    (Maclaurin.) 


CHAPTER  III 

FUNCTIONS  OF  TWO  VARIABLES 
I.    ORDINARY  EXTREMES 

16.  Let  z=f(x,  y)  be  a  continuous  function  of  the  two  vari 
ables  x  and  y  when  the  point  P  with  coordinates  (x,  y)  remains 
within  the  interior  of  an  area  H  which  is  limited  by  a  contour  C. 
We  say  that  this  function  f(x,  y)  is  a  minimum  for  a  point  (XQ,  yQ) 
of  the  area  fl  when  we  can  find  a  positive  quantity  B  such  that 

we  have  A  =  /(,,0+  h,  y, +  *)-/(*„  y0)  &  0  (i) 

for  all  systems  of  values  of  the  increments  h  and  k  that  are 
less  than  8  in  absolute  value.  The  maximum  is  defined  in  a 
similar  manner.* 

If  we  exclude  the  sign  =  in  the  expressions  A  ^  0  or  A  ^  0, 
the  extremes  are  said  to  be  proper  (cf.  §  1);  but  if  the  equality 
A  =  0  exists  for  certain  values  of  h  and  k  that  are  less  than 
8  in  absolute  value,  however  small  8  be  taken,  we  have  improper 
extremes.  For  example,  in  the  case  of  the  surface  represented 
by  the  equation  z  =/(#,  y},  the  axis  Oz  being  vertical,  a  proper 
maximum  corresponds  to  an  isolated  summit,  but  if  these  sum 
mits  form  a  line  on  the  surface,  this  line  will  be  a  line  of 
improper  maxima.  Consider,  for  example,  the  lines  generated  by 
revolving  the  extremes  of  a  plane  curve  about  the  Ox-axis. 

If  in  the  expression  (/)  we  regard  y  as  constant  and  equal 
to  y0,  then  z  becomes  a  function  of  one  variable  x  and  (§  2) 
the  difference  /.,  .  7  x  /.,  N 

/(*„+*>  y«)— /K»yo) 

can  only  retain  a  constant    sign    for   small   values   of  h  if   the 

O   f 

derivative  ~  is  zero  for  x  =  xQt  y  =  y0. 

ex 

*  See  also  Goursat,  loc.  cit. 
23 


24  THEORY  OF  MAXIMA  AND  MINIMA 

In  the  same  way  it   may  be  shown  that  these  values  must 

also  cause  —  to  be  zero.    It  follows  that  the  systems  of  values 
dy 

which  cause  f(x,  y)  to  become  proper  extremes  are  to  be  found 
among  the  solutions  of  the  two  simultaneous  equations 


' 

dx  dy 

conditions  which  are  also  necessary  for  improper  extremes. 

As  only  ordinary  extremes  are  considered  here,  the  partial  deriva 
tives  of  the  second  order  of  f(x,  y)  are  supposed  to  be  continuous 
(§  11)  in  the  neighborhood  of  the  values  XQ,  yQ  and  are  not  all  zero 
for  x0,  yQ,  and,  furthermore,  the  derivatives  of  the  third  order  are 
supposed  to  exist.  If,  then,  x  =  x0  and  y=yQ  are  a  solution  of  the 
two  equations  (a),  the  formula  for  Taylor's  theorem  gives  us 


For  values  of  h  and  k  in  the  neighborhood  of  zero,  it  is  clear  that 
the  trinomial 


gives  its  sign  to  the  right-hand  side  of  (ii),  and  it  is  evident 
that  the  discussion  of  the  sign  of  this  trinomial  is  going  to 
enjoy  a  preponderant  role. 

To  have  an  extreme  for  x  =  XQ,  y  =  y^  it  is  necessary  and 
sufficient  that  the  difference  A  retain  a  constant  sign  when  the 
point  (xQ  +  h,  yQ  +  k)  remains  within  the  interior  of  a  square 
sufficiently  small  which  has  the  point  (XQ,  yQ)  for  center.  In 
this  case  the  difference  A  will  also  retain  a  constant  sign  if 
the  point  (x0+h,  yQ+  k)  remains  within  a  circle  with  radius 
sufficiently  small  and  center  (XQ)  yQ),  and  inversely  ;  for  we  may 
replace  the  square  by  the  inscribed  circle  and  reciprocally. 
Suppose,  then,  that  C  is  a  circle  of  radius  r  with  the  point 


FUNCTIONS  OF  TWO  VAEIABLES  25 

(XQ,  y0)  as  center.    We  have  all  the  interior  points  of  this  circle 
by  writing  h  =  p  cos  <£,  k  =  p  sin  (f>  and  causing  (/>  to  vary  from 
0  to  2  TT  and  by  causing  p  to  vary  from   —  r  to  4-  r. 
Making  this  substitution  in  A,  it  becomes 

A  =  f^cos24>  +  2.Ssin</>  cos  c/>  +  (7sin2</>)  +  ^  A 

21  <J  : 

where  yl  =  -~  >  ^  =  TT— ^— »  C  =  ^r  >  and  where  L  is  an  expres- 


cx*  dxQ2y0  dy 


sion  which  retains  a  finite  value  in  the  neighborhood  of  the 
point  (a?0,  y0). 

It  is  evident  that  several  cases  are  to  be  distinguished  according 
to  the  sign  of  &  —  AC. 

17.  First  case.    B*-AOQ. 

The  equation  A  cos2<f>  +  2  B  sin  <f>  cos  <f>  -h  (7  sin2<£  =  0  admits 
two  real  roots  in  tan  $,  and  the  left-hand  side  may  be  written  as 
the  difference  of  two  squares  in  the  form 


A  =      [a  (a  cos  <£  +  &  sin  </>)-  /(acos  </>  +     sn 
where  «>0,     yS>0,     and     (ab'-ba')^Q. 

By  takmg  the  circle  sufficiently  small  we  may  neglect  the 
terms  of  the  third  and  higher  degrees  in  p.  If  next  to  the  angle  </> 
a  value  is  given  such  that  a  cos  <f>  +  b  sin  <f>  =  0,  it  is  seen  that  A 
will  be  negative  ;  while  if  we  give  the  angle  <f>  a  value  such  that 
a'cos<^  +  ft'sin^  =  0,  then  A  will  be  positive. 

It  is  therefore  impossible  to  find  a  number  r  such  that  the  dif 
ference  A  retains  a  constant  sign  when  the  absolute  value  of  p  is 
inferior  to  r,  while  the  angle  <£>  is  arbitrary.  It  follows  that  the 
function  f(x,  y)  has  no  extreme  for  x  =  xQt  y  =  yQ. 

18.   Second  case.    B2-AC<0. 

It  is  evident  that  A  and  C  must  have  the  same  sign. 

The  trinomial 

A  cos2<£  +  2  B  cos  (f>  sin  $  4-  C  sin2<£ 

=  -  [(A  cos<f>+B  sin  <#>)2  +  (AC  -  JS^sin2^] 

y± 

does  not  vanish  when  <$>  varies  from  0  to  2  TT. 


26  THEORY  OF  MAXIMA  AND  MINIMA 

Let  w  be  the  lower  limit  of  the  absolute  value  of  the  trinomial 
and  let  H  be  the  upper  limit  of  the  absolute  value  of  the  function 
L  in  a  circle  of  radius  R  and  center  (xn,  yn). 

\        0>       9W  Q 

Let  r  be  a  positive  number  inferior  to  R  and  to  --    Within 

H 
the  circle  of  radius  r  the  difference  A  will  have  the  same  sign  as 

the  coefficient  of  /a2,  that  is  to  say,  of  A  or  C.    The  function  /(#,  y) 
has  therefore  an  extreme  for  x  =  #0,  y  =  y0. 

19.  The  above  results  may  be  summarized  as  follows:  If  at 
the  point  #0,  y0  we  have 

ay       ay  ay 


there  is  wo  extreme  ;  but  if 

/  gy  Y    ay  ay 
^Vy0/    az»  ay* 

there  is  a  maximum  or  minimum  according  to  the  sign  of  the  two 

,   .    ..      ay   a2/ 

derivatives  —  ^-  >  —  ^-  • 


There  is  a  maximum  if  these  derivatives  are  negative,  a 

if  they  are  positive,  and  it  is  also  seen  that  we  have   a 


proper  maximum  or  minimum.* 

Example.    In  the  theory  of  least  squares  it  is  required  to  determine  x,  y 
so  that  the  expression 

(A}  u  (x,  y)  =  V  (akx  +  bky  +  c*)2 

k  =  l 

may  be  as  small  as  possible.    In  other  words,  determine  the  values  of  x  and 
y  for  which  u  (x,  y)  is  equal  to  its  lower  limit. 

Following  the  methods  indicated  above  we  must  solve  the  two  equations 


Ft  is  seen  that  the  determinant  of  these  equations  is  equal  to  the  sum  of  the 
i  n  (n  —  1)  squares  (a^  —  rt$fc)a(&,  Z  =  1,  2,  •  •  •,  n ;  k  <  I),  and  this  deter 
minant  does  no^  vanish  if  among  the  binomials  a#r  +  lky  there  are  at  least 

*  See  Lagrange,  Misc.  Taur.,  Vol.  I.    1759. 


FUNCTIONS  OF  TWO  VAKIABLES  27 

two  which  do  not  differ  from  each  other  by  a  constant  factor.  Under  this 
assumption  the  two  equations  (Z?)  have  one  and  only  one  system  of  solu 
tions  x0,  yQ. 

That  «  does  in  fact  reach  its  lower  limit  for  this  pair  of  values  is  seen 
if  we  write  in  (A)  x  =  XQ  +  £,  y  —  y0  +  r],  and  expand.  We  then  have 

k=n 
u  (x0  +  £,  y0  +  77)  —  u  (JTO,  y0)  -  V  («jt£  +  &A-7/)2, 

k  =  l 

which  difference  is  a  positive  quantity  for  every  system  of  values  except 
£  =  0,  77  =  0. 

PROBLEMS 

1.  Find  a  point  P  of  a  plane  such  that  the  sum  PA  +  PB  +  PC  of  its 
distances  to  three  fixed  points  of  the  plane  is  a  minimum.    In  particular 
consider  the  case  when  BA  C  >  120°,  and  show  that  here  the  point  A  gives 
the  minimum.    (Cavalieri,  Exercitationes  Geometricae,  pp.  504-510.    1647.) 

2.  In  a  plane  triangle  all  of  the  angles  have  been  measured  with  the 
same   precision   and  found  to  have  values  a,  (3,  y.    On  account  of  the 
unavoidable  error  in  observation,  the  sum  a  +  )8  +  y  does  not  equal  180°. 
Let  the  difference  180  —  (a  +  /3  +  y)  be  equal  to  S,  where  8  is  expressed  in 
circular  measure.     What  values  u,   v,  w   (in  circular  measure)  must  be 
added  to  the  three  results  of  measurement  if  we  wish 

(1)  that  a  +  /3  +  y  +  u  +  v  +  w  =  180,  and 

(2)  that  u2  +  r2  +  ic'2  be  as  small  as  possible  ? 
Answer,    u  =  ^  8  =  v  =  tc. 

INTRODUCTION  TO  THE  AMBIGUOUS  CASE  B2—AC=Q 

20.  We  shall  first  note  the  difficulties  that  attend  this  special 
case,  and  with  Goursat*  we  shall  illustrate  these  difficulties  by 
means  of  geometric  considerations ;  we  shall  then  call  attention  to 
erroneous  deductions  which  have  been  made,  and  later  a  method 
will  be  given,  due  to  Scheeffer,  of  determining  the  extremes  for 
this  case,  when  they  exist. 

Let  S  be  the  surface  represented  by  the  equation  z=f(x,y). 
If  the  function  f(x,  y)  has  an  extreme  at  the  point  XQ,  T/O,  and  if 
the  function  and  its  derivatives  are  continuous,  we  must  have 


*  Goursat,  p.  112. 


28  THEORY  OF  MAXIMA  AND  MINIMA 

which  shows  that  the  tangential  plane  to  the  surface  S  at  the 
point  PQ  (with  coordinates  XQ)  y0,  ZQ)  must  be  parallel  to  the 
^y-plane.  In.  order  that  this  point  shall  correspond  to  an  ex 
treme,  it  is  necessary  that  in  the  neighborhood  of  the  point  P0 
the  surface  S  be  entirely  on  one  side  of  the  tangential  plane.  We 
are  thus  led  to  the  study  of  a  surface  with  regard  to  a  tangential 
plane  in  the  neighborhood  of  the  point  of  contact. 

Suppose  that  the  origin  has  been  transposed  to  the  point  of 
contact.  The  tangential  plane  being  taken  as  the  a^-plane,  the 
equation  of  the  surface  is  of  the  form 

z  =  ax2  -f-  2  bxy  -f-  cy2  +  ay?  -f-  3  fix^y  -f-  3  yxy2  +  £?/3,  (i) 

where  &,  b,  c  are  constants  and  #,  /3,  7,  8  are  functions  of  xt  y 
which  remain  finite  when  x  and  y  tend  towards  zero.  To  deter 
mine  whether  the  surface  S  is  situated  entirely  on  one  side  of 
the  a?2/-plane  in  the  neighborhood  of  the  origin,  it  is  sufficient 
to  study  the  intersection  of  this  surface  by  the  ajy-plane.  This 
intersection  is  a  curve  C  represented  by  the  equation 

f(x,  y)  —  ax2  +  2  Ixy  -f  c?/2-f-  ax*+  .  .  .  =  0,  (ii) 

and  presents  a  double  point  at  the  origin. 
21 .  If.  &2  —  ac  is  positive,  the  equation 

ax*+  2  Ixy  +  cy*=-  [(ax  +  %)2-  (52-  ac) 7/2]  =  0 

Ch 

represents  two  real  and  distinct  straight  lines  which  pass  through 
the  origin.  Suppose  that  we  take  these  two  lines  for  the  axes 
of  coordinates,  and  note  that  this  is  brought  about  by  a  linear 
change  of  the  variables. 

The  equation  (ii)  then  has  the  form 

xy+R(x,y)=Q.  (Hi) 

If  in  this  equation  we  write  y  =  ux,  we  have 

R  (X,  U3$)  . .  . 

»=-.  ^-   '  (•") 

where  it  is  evident  that  R(x}  ux)  is  divisible  by  x*. 


FUNCTIONS  OF  TWO  VARIABLES 


29 


It  follows  from  §  140  (see  also  Goursat,  §  34)  that  equation 
(iv)  has  one  and  only  one  root,  say  u  =  ?(#),  which  tends  towards 
zero  with  x.  Hence  through  the  origin  there  passes  one  branch 
of  the  curve  C  represented  by  an  equation  y  =  #£(#),  which  is 
tangent  at  the  origin  to  the  axis  Ox.  If  we  interchange  the 
role  of  the  two  variables  x  and  yt  it  is  seen  that  there  also 
passes  through  the  origin  a  second  branch  of  the  curve  C  which 
is  tangent  to  the  axis  Oy.  The  point  O  is  a  double-point  with 
distinct  tangents. 

If,  then,  62—  «e  >  0,  the  intersection  of  the  surface  S  by  the 
tangential  plane  presents  two  distinct  branches  of  curve  Cl  and 
<72  which  pass  through  the  origin,  and  the  tangents  to  these  two 
branches  of  curve  at  the  origin  are 
represented  through  the  equation 


FIG.  7 


If  we  give  to  each  region  of  the 
plane  in  the  neighborhood  of  the 
origin  the  sign  of  the  first  term  in 
(iii)}  as  seen  in  the  figure,  it  is  clear 
that  if  a  point  moves  along  either 
of  the  curves  Cl  or  <72,  the  left-hand 
side  of  (iii)y  and  consequently  also 
of  (ii),  changes  sign  as  the  point  passes  through  the  origin.  It  fol 
lows  that  f(x,  y)  does  not  have  an  extreme  (cf.  §  17)  at  the  origin. 

22.  If  b2  —  ae<0,  the  origin  is  a  double  isolated  point  ;  for 
within  the  ulterior  of  a  circle  with  sufficiently  small  radius 
described  about  the  origin  as  center,  the  right-hand  side  of  (ii) 
only  vanishes  at  the  origin  itself.  To  show  this  write  x  =  p  cos  <f>, 
y  =  r  sin  <£,  where  x  and  y  are  the  coordinates  of  a  point  in  the 
neighborhood  of  the  origin. 

Equation  (ii)  becomes 

f(xt  y}  =  p2(a  cos2(£  +  2  b  sin  <f>  cos  <£  +  c  sin2<£  +  pL) 

where  L  is  a  function  of  p  and  <£  which  remains  finite  when  p 
tends  towards  zero.  Let  H  be  the  upper  limit  of  \L\  when  p  is  less 
than  a  positive  number  r. 


30  THEORY  OF  MAXIMA  AND  MINIMA 

When  <f)  varies  from  0  to  2  TT;  the  trinomial 

a  cos2</>  +  2  b  sin  (f>  cos  (/>  +  c  sin2<£ 

retains  a  constant  sign.  Let  m  be  the  minimum  of  its  absolute 
value.  It  is  clear  that  the  coefficient  of  p2  does  not  vanish  for 
any  point  on  the  interior  of  a  circle  C  with  radius  less  than  r  and 

777 

—  »  having   the    origin    as    center.     Consequently    the    equation 

/(a?,  y)  =  0  admits  of  no  other  solution  than  x  =  0,  y  =  0  (i.e.,  p  =  0) 
within  the  circle. 

It  follows  that  f(x,  y)  retains  a  constant  sign  when  the  point  x,  y 
moves  within  the  interior  of  this  circle.  Hence,  also,  all  the  points 
excepting  the  origin  of  the  surface  S  which  may  be  projected  upon 
the  circle  C  are  situated  upon  the  same  side  of  the  ^cy-plane.  The 
function  /(a;,  y)t  therefore,  presents  an  extreme  at  the  origin  (cf.  §  18). 

23.  When  62—  ac  =  0,  the  two  tangents  at  the  double  point 
coincide,  and  there  are,  in  general,  two  branches  of  curve  tangent 
to  the  same  straight  line,  which  form  a  cusp. 

The  complete  study  of  this  theory  will  be  found  to  require  a 
somewhat  delicate  discussion. 

For  example,  y2=x?  presents  at  the  origin  a  cusp  of  the  first 
kind  ;  that  is,  one  which  has  the  two  branches  of  curve  tangent  to 
the  Oo>axis  lying  the  one  above  and  the  other  below  this  tangent. 

The  curve  y*—  2  x*y  -f  zt—  ofi=  0  presents  a  cusp  of  the  second 
kind  ;  the  two  branches  of  curve  are  tangent  to  the  a?-axis  and 
situated  on  the  same  side  of  it.  The  equation  gives  us,  in  fact, 
y  =  x2  ±  x*.  The  two  values  of  y  have  the  same  sign  in  the  neigh 
borhood  of  the  origin  and  are  only  real  when  x  is  positive. 

The  curve  ^+  x*y*—  6  x*y  -f  y*=  0  presents  two  branches  of 
curve  which  offer  nothing  peculiar,  both  being  tangent  at  the  origin 
to  the  #-axis.  We  have  from  this  equation 


from  which  it  is  seen  that  the  two  branches  obtained  when  we 
take  successively  the  two  signs  before  the  radical  have  no  singu 
larity  at  the  origin. 


FUNCTIONS  OF  TWO  VARIABLES  31 

It  may  also  happen  that  the  curve  is  composed  of  two  coincident 
branches,  as  is  the  case  of  the  curve  represented  by  the  equation 

f(xt  ?/)  =  y2-  2  o%  +  ^=  0  ;    that  is,  (y  -  £2)2  =  0. 

It  is  evident  that  here  the  left-hand   side   passes  through  zero 
without  changing  sign. 

It  may  also  occur  that  the  point  #0,  yQ  is  a  double  isolated  point, 
as  is  presented  through  the  curve 


at  the  origin. 

From  the  above  it  is  seen  that  if  the  origin  is  a  double  isolated 
point,  or  if  the  intersection  of  the  surface  with  the  tangent  plane 
is  composed  of  two  coincident  branches,  the  function  f(x,  y)  will 
be  an  extreme  (hi  the  latter  case  just  given  an  improper  extreme); 
but  if  the  intersection  is  composed  of  two  distinct  branches  which 
pass  through  the  origin,  there  will,  in  general,  be  no  extreme,  for 
the  surface  again  cuts  its  tangential  plane. 

24.  Take,  for  example,*  the  surface 


which  cuts  its  tangential  plane  along  two  parabolas  of  which  the 
one  is  interior  to  the  other.  That  the  surface  may  not  cross  its 
tangential  plane,  it  is  necessary  that  if  we  cut  this  surface  by  any 
cylinder  having  its  elements  parallel  to  Oz  and  passing  through  Oz, 
the  curve  of  intersection  shall  lie  on  one  side  of  the  #?/-plane. 

Let  y  =  (f)  (x)  be  the  trace  of  such  a  cylinder  upon  the  xy- 
plane,  the  function  <j>(x)  being  zero  for  x  =  0.  If  /(O,  0)  is  to  be 
a  minimum,  the  function  f(x,  <f>  (x))  =  F(x),  say,  ought  to  be  a 
minimum  for  x  —  0,  whatever  the  function  <j>  (x). 

To  simplify  the  calculation,  suppose  that  we  have  chosen  the 
axes  of  coordinates  so  that  the  equation  of  the  surface  is  of  the 
form  . 


where  A  is  a  positive  quantity. 

*  This  is  a  generalization  due  to  Goursat  (p.  llo)  of  the  classic  example  of  Peano 
(loc.  cit.,  Nos.  133-136). 


32  THEORY  OF  MAXIMA  AND  MINIMA 

With  this  system  of  axes  we  have  for  the  origin 


The  derivatives  of  ^(a?)  are 


For  x  =  0  =  y  these  formulas  become 


If  </>'(0)  =£  0,  the  function  J^(aj)  evidently  has  a  minimum  for 
x  =  0  ;  but  if  0'(0)  =  0,  it  is  seen  that 


and 


Hence,  in  order  that  F(x)  be  a  minimum,  it  is  necessary  that 

3 

be  zero,  while 


dx* 


must  be  positive,  whatever  the  value  of  <£"(()). 
These  conditions  are  not  satisfied  for  the  surface 


considered  above,  while  they  are  satisfied  for  the  function 
.    z  =  ?/2  4-  #*• 

It  is  thus  seen  that  in  the  ambiguous  case,  where  B*  —  A  C  =  0, 
the  derivation  of  the  necessary  and  sufficient  conditions  for 
the  extremes  of  functions  of  only  two  variables  is  going  to  be 
accompanied  by  difficulties.  It  is  also  evident  that  in  the  case 
of  three  or  more  variables  these  difficulties  will  be  correspond 
ingly  augmented. 


FUNCTIONS  OF  TWO  VARIABLES  33 

II.   INCORRECTNESS  OF  DEDUCTIONS  MADE  BY  EARLIER 
AND  MANY  MODERN  WRITERS 

25.  One  of  the  greatest  mathematicians  of  all  times,  Lagrange 
(Theorie  des  Fonctions,  p.  290),  writes: 

If  all  the  terms  of  the  first  and  second  dimensions  [see  formula  (7/)  of 
§  16]  vanish,  it  is  necessary  for  the  existence  of  a  maximum  or  minimum 
that  all  the  terms  of  the  third  dimension  in  hv  7*2,  •  •  •  shall  disappear  and 
that  the  quantity  composed  of  terms  where  hv  hz,  •  •  •  (cf.  §  51)  form  four 
dimensions  shall  be  always  positive  for  the  minimum  and  always  negative 
for  the  maximum  when  fiv  /?„,  •  •  •  have  any  values  whatever. 

Following  Lagrange,  all  writers  on  this  subject  made  the  same 
incorrect  deductions  until  Peano,  in  the  remarks  to  Nos.  133-136 
found  in  the  Appendix  to  his  Calcolo,  wrote :  "  The  proofs  for 
the  criteria  by  which  the  maxima  and  minima  of  functions  of 
several  variables  are  to  be  recognized,  and  which  are  given  in 
most  books,  depend  upon  the  theorem  that  in  the  Taylor  develop 
ment  for  functions  of  several  variables  the  ratio  of  the  remainder 
after  an  arbitrary  term  to  this  term  has  a  limit  zero  when  the 
increments  of  the  variables  approach  zero.  This  theorem  is  in 
general  false  when  the  term  in  question  is  not  a  definite  form 
with  respect  to  the  increments  of  the  variables,  and  when  it  is  a 
definite  form,  the  theorem  needs  proof." 

These  fallacious  conclusions  are  found,  for  example,  in  Bertrand 
(Calciil  Differentiel,  p.  504),  and  also  in  Serret  (Calc.,  p.  219), 
who  writes: 

The  maxima  or  minima  exist  if  for  the  values  hv  A2,  •  •  •  which  cause  d?f 
and  dPf  to  vanish  the  derivative  d*f  has  invariably  the  minus  or  plus  sign. 

Here  d2/,  c?3/,  •  •  •  denote  the  homogeneous  integral  forms  of 
the  second,  third,  •  •  •  degrees  in  hlt  A2,  •  •  •,  when  the  function  / 
is  expanded  by  Taylor's  theorem  (cf.  §  51). 

Todhunter  (pp.  227-229  of  the  1864  and  1881  editions  of 
his  Calculus),  for  the  semi-definite  case  where  B2—AC=  0,  writes 
the  Taylor  expansion  for  a  function  of  two  variables  in  the  form 
(see  («)  of  §  16)  ,2  ,  ,  X2 


where  R±  is  the  remainder  term. 


34 


THEORY  OF  MAXIMA  AND  MINIMA 


The  condition  which  it  appears  that  he  considered  as  suffi 
cient  for  an  extreme  is  that  A  and  R±  must  have  the  same 
sign,  and  if  the  terms  of  the  second  dimension  are  zero  for 
the  position  or  positions  in  question,  then  also  the  terms  of 
the  third  dimension  must  be  zero. 

That  this  is  not  true  is  seen  at  once  by  observing  Peano's 
classic  example  ^  y)  =  (y-  P2x2)  (y  -  q2x2), 

where  the  conditions  just  mentioned  exist,  although  there  is  no 
extreme  at  the  origin,  as  already  seen  in  §  24. 

Professor  Pierpont  (Bull,  of  the  Am.  Math.  Soc.,  Vol.  IV,  p.  536) 
says,  "  Our  English  and  American  authors  seem  to  be  ignorant 
of  these  facts." 

Write  Peano's  example  in  the  form 


It  is  seen  that  the  function  /(#,  y)  is  positive  in  the  neighbor 
hood  of  the  origin  upon  every  straight  line  through  it;  however, 
upon  the  parabola  y  =  mx2  the  function  in  the  neighborhood  of 
the  origin  is  positive,  zero, 
or  negative  according  as 
am2  +  2  bm  +  c  is  positive, 
zero,  or  negative. 

We  may  further  illustrate 
this  as  follows  :  Let  the 
equations 


denote  two  curves  through 
the  origin.    The  function 

Fir    8 

will  have  positive  values  for  values  of  x,  y  on  the  arc  BA  of  a  circle 
with  origin  at  the  center  and  radius  sufficiently  small  and  negative 
values  on  the  arc  AD.  Hence  the  function  f(x,  y)  has  minimum 
values  on  all  straight  lines  through  the  origin  that  cut  the  arcBA  and 
maximum  values  on  the  lines  through  the  origin  that  cut  the  arc  AD. 


FUNCTIONS  OF  TWO  VARIABLES 


35 


If,  further,  the  two  curves  4>  (x,  y]  =  0,  NP  (a;,  y)  =  0  have  a  com 
mon  tangent  at  the  origin  with  their  curvatures  lying  in  the  same 
direction,  it  is  seen  that  all  possible  straight  lines  through  the 
origin  are  such  that  the  coordi-  Y 

nates  of  any  points  on  them  cause 
f(x,  y)  to  have  positive  values. 
This  is  true,  for  example,  of  the 
function  already  considered, 


f>o 


FIG.  9 


In  the  spaces  above  and  below 
both  curves  we  have/(#,  y)  >  0, 
while  this  function  is  negative 
for  the  spaces  between  the  two 
curves;  so  that  there  is  a  minimum  upon  every  straight  line 
through  the  origin,  although  there  is  a  maxiumm  *  of  f(x,  y)  for 

all  points  on  the  curve  y  = 


III.    DIFFERENT  ATTEMPTS  TO  IMPROVE  THE  THEORY 

26.  The  existence  of  an  extreme  of  the  function  f(x,  y)  at  the 
origin,  for  example,  a  minimum,  depends  upon  the  condition  that 
there  exists  an  upper  limit  g  such  that  the  function  f(xt  y)  for  all 
values  of  xy  y  which  satisfy  the  condition 


is  positive  ;  or,  geometrically  speaking  (§  16),  this  condition  implies 
that  there  exists  a  circle  with  center  (0,  0)  within  which  the  func 
tion  is  everywhere  positive  with  the  exception  of  the  position 
(0,  0)  itself. 

Instead  of  considering  the  values  of  such  a  function  for  the 
coordinates  of  points  on  straight  lines  through  the  origin,  which 
lines  may  be  written  in  the  form 

x  =  ak,         y  =  flk, 


*  Note  in  this  connection  Scheeffer,  Math.  Ann.,Vo\.  XXVI,  p.  197 ;  and  Vol.  XXXV, 
p.  545. 


S6  THEORY  OF  MAXIMA  AND  MINIMA 

a,  ft  being  arbitrary  constants,  it  would  be  natural  to  raise  the 
question  whether  we  could  not  determine  the  sufficient  conditions 
for  such  extremes  by  studying  the  more  general  curves  expressed 
through  the  algebraic  equations 


x  (k)  =  a^k  +  aj£  -\  ---- 


and  make  the  requirement  that  the  function  f(x(k),  y(k})  shall 
have  an  extreme  for  k  =  0,  whatever  values  there  may  be  assigned 
to  the  positive  integers  m  and  n  and  to  the  m  +  n  quantities  alt 
a2>  '  '  •>  am>  A»  @2>  '  '  '>  Pn>  &  being  of  course  assumed  that  all 
the  quantities  a  and  /3  are  not  simultaneously  zero. 

It  may,  however,  be  shown  that  such  sufficient  conditions  cannot 
be  derived  in  the  manner  indicated.  For  if  we  write 

4>  (x,  y)  =  y  —  sm2x, 
ip(x,  y)  =  y  —  sin2#  —  e  x*t 

we  have  two  curves  denned  through  the  equations  <E>  (x,  y)  =  0 
and  M?  (x,  y)  =  0  which  have  at  the  origin  the  a?-axis  as  a  common 
tangent  and  a  contact  of  an  indefinitely  high  order. 

There  is  consequently  no  curve  of  the  form  (i)  which  may  be 
laid  between  these  two  curves  ;  for  clearly  any  such  curve  must 
have  with  either  of  these  curves  a  contact  of  indefinitely  high 
order  which  is  impossible  for  an  algebraic  curve. 

On  the  other  hand,  the  function  f(x,  y)  =  <S>  (x,  y)  M*  (x,  y)  is 
positive  in  the  whole  plane  excepting  that  part  of  the  plane  that 
is  situated  between  the  two  transcendental  curves,  in  which  it  is 
negative.  Hence  at  the  origin  there  is  neither  a  maximum  nor  a 
minimum  for  the  function  f(x,  y)t  although  for  this  function  upon 
every  curve  (i)  there  enters  a  minimum. 

We  may  therefore  desist  from  further  requirements  in  this 
direction,  and  we  shall  next  call  attention  to  two  methods,  the 
one  due  to  Scheeffer  and  the  other  to  Von  Dantscher,  which  are 
general  in  character  when  the  discussion  has  to  do  with  two 
variables  and  which  lead  to  criteria  which  are  of  use  in  practice. 


FUNCTIONS  OF  TWO  VARIABLES  37 

27.  Scheeffer's  method.  We  have  seen  that  functions  of  one 
variable  which  have  ordinary  extremes  can  be  expressed  through 
the  Taylor  development  in  the  form 

/(,;)  =  /!!(M^     (0<0<1),  (a) 

71  I 

when  f(x)  and  the  derivatives  f'(x),  •  •  -,  f(n~l\x)  are  zero  for 
x  =  0  while  f(n\x)  =£  0  for  x  =  0.  For  such  functions  the  change 
in  value  in  the  neighborhood  of  the  position  x  =  0  on  either 
side  is  faster  than  that  of  a  given  quantity  axn  ;  that  is,  positive 
quantities  a,  n,  and  S  may  be  so  chosen  that  for  all  values  of  x 
within  the  interval  —  S  to  +  S  the  absolute  value  of  f(x)  is 
greater  than  the  absolute  value  of  axn,  excepting  the  value  x  =  0. 
For  since  /(n)(0)  =£  0,  we  may  so  determine  8  that  for  values  of 
x  such  that  —  8  ^  x  ^  8  the  function  /(n)(#)  is  different  from 
zero.  If,  then,  we  choose  the  quantity  ci  smaller  than  the  absolute 

f(n)/x\ 

value  of  -  —  *-£  in  the  interval  —  8  to  +  8,  then  (see  formula  (a)) 

n  i 

within  this  interval  the  condition  |/(<£)|  >  axn\  is  satisfied.  Recip 
rocally,  if  the  last  condition  exists,  the  n  first  derivatives  of 
f(x)  cannot  all  vanish  for  x  =  0.  For  in  the  latter  case  we 
would  have  (*+V 


(n  -hi)! 
and  from  this  it  follows  that 

Em  £8  =  0 


which  contradicts  the  assumption  that  |/(#)|  >  \axn\. 

There   are   functions,  however,  for   example  e  x"  (cf.  Pierpont, 

loc.  cit.,  Vol.  I,  p.  205),  for  which  such  quantities  n,  a,  8,  do  not 

_^ 

exist.    In   fact,  the   absolute  value  of  e  •*•*  is  in  the  immediate 
neighborhood  of  x  ='0  smaller  than  any  arbitrary  power  axn. 

We  may  note  that  the  characteristic  property  of  the  above 
requirement  consists  in  the  fact  that  the  behavior  of  the  function 
in  the  neighborhood  of  the  origin  must  be  marked  with  a  certain 
degree  of  distinctness. 


38  THEORY  OF  MAXIMA  AND  MINIMA 

The  following  consideration  leads  to  the  generalization  of  the 
above  condition  for  functions  of  two  variables:  It  is  clear  that  a 
function  f(x,  y)  which  vanishes  at  the  origin,  if  it  is  continuous, 
has  upon  the  circumference  of  every  circle  which  is  described 
about  the  origin  as  center  with  an  arbitrary  radius  r  a  greatest 
and  a  least  value,  provided  the  function  does  not  reduce  to  a 
function  of  one  variable  r  =  V  y?  -f-  z/2.  The  signs  of  these  greatest 
and  least  values,  which  we  denote  by  /x  (r)  and  /2  (r)  respectively, 
offer  for  sufficiently  small  radii  r  a  criterion  regarding  the  appear 
ance  or  nonappearance  of  an  extreme  at  the  origin.*  For  if  the 
two  quantities  /x  (r)  and  /2  (r)  are  positive,  there  will  be  a  mini 
mum  of  f(x,  y)  at  the  origin,  while  if  they  are  both  negative,  a 
maximum  exists  at  the  origin.  The  degree  of  distinctness  which 
marks  the  behavior  of  the  function  at  the  origin  is  characterized 
through  the  existence  of  a  power  arn  with  the  property  that  for 
every  value  of  r  within  a  certain  limit  g  both  f1  (r)  and  /2  (r) 
are  in  absolute  value  greater  than  ar11. 

If  this  requirement  is  not  satisfied  we  cannot  count  upon 
deriving  sure  characteristics  of  extremes  through  the  expansion 
in  series.  For  in  this  case  the  value  with  which  the  function 
f(x,  y)  in  the  neighborhood  of  the  position  (0,  0)  either  ap 
proaches  the  value  zero  from  the  one  side,  or  having  passed 
through  zero  differs  from  it  on  the  other  side,  is  so  little  that 
this  value  cannot  be  expressed  through  a  power  ever  so  high 
of  r.  The  development  in  series  cannot,  therefore,  serve  to  de 
termine  whether  the  value  is  a  little  on  the  one  side  or  on  the 
other  side  of  zero. 

As  examples  of  this  kind  are  the  function 


which  has  a  minimum  value  at  the  origin,  and  the  function 


*  The  behavior  of  the  function  f(x,  y)  at  any  point  JBO,  yQ  other  than  the  origin 
may  be  made  by  the  substitution  x  =  x0  -f  h,  y  —  yQ  +  k,  to  depend  upon  the  behavior 
of  the  function  /(KO  +  h,  yQ  +  k)  =  F  (h,  k)  for  the  values  h  —  0,  k  =  0. 


FUNCTIONS  OF  TWO  VARIABLES  39 

which  has  neither  a  maximum  nor  minimum  at  the  origin.    The 
first  function  approaches  the  value  zero  from  the  positive  direction 

_£! 
up  to  the  value  e   ^(for  y=  0)  while  the  latter  approaches  the  value 

zero  from  the  negative  direction  by  the  same  amount. 

To  this  class  of  functions  belong  also  those  functions  whose 
initial  terms  constitute  a  semi-definite  form  and  which  contain  as 
a  factor  an  even  power  of  a  series  P(x,  y)  the  terms  of  which 
vanish  for  real  pairs  of  values  x,  y  in  every  region  arbitrarily 
small  where  0  <  \x\  <  8,  0  <  \y  <8  (see  §§  36  and  41).  Belonging 
also  to  this  category  of  functions  are  the  functions  which  reach 
the  value  zero  but  do  not  pass  through  it  for  every  region  arbi 
trarily  small  where  0  <  x\<8,  Q<\y\<8. 

If  on  the  other  hand  there  exists  a  power  arn  whose  value,  so 
long  as  we  remain  within  a  certain  limit  g,  is  always  smaller  than 
the  absolute  values  of  fi(r)  and/2(r),  then  the  question  whether 
at  the  origin  an  extreme  of  the  function  exists  may  always 
be  answered  by  a  development  in  series  and  by  a  finite  num 
ber  of  observations.  How  this  is  accomplished  is  found  in  the 
next  chapter. 

28.  The  method  of  Von    Dantscher.    We   have   seen  that  by 
considering  the  extremes  on  every  line  through  (0,  0)  we  are  not 
able  to  form  any  conclusions  regarding  the  extremes  of  the  func 
tion  f(x,  y)  at  this  point.    Von  Dantscher's  method  consists  in 
establishing  criteria  not  only  for  the  extremes  on  such  lines  but 
also  for  all  points  in  the  plane  in  the  neighborhood  of  the  points 
on  these  lines  and  also  in  the  neighborhood  and  on  both  sides  of 
the  point  (0,  0).   Although  Von  Dantscher  himself  finds  that  there 
is  "no  need  of  an  extension  or   improvement    of   the   Scheeffer 
method,"  I  shall  give  later  the  method  of  Von  Dantscher,  as  it  is 
of  interest  in  itself  and,  besides,  it  is  well  to  compare  the  two 
theories  (see  §§  42,  44). 

29.  The  Stolzian  theorems.*   Wre  shall  at  first  assume  that  the 
function  f(x,  y)  is  continuous  with  respect  to  both  variables  in 
every  point  (x,  y}  of  a  rectangle  that  includes  the  point  (0,  0),  the 

*  Stolz,  p.  213. 


40 


THEORY  OF  MAXIMA  AND  MINIMA 


sides  of  the  rectangle  being  parallel  to  the  coordinate  axes.   We 
shall  state  and  then  prove  the  following  theorems  : 

THEOREM  I.  A  necessary  condition  that  f(0,  0)  be  a  proper 
extreme  of  f(x}  y}  is  offered  through  the  existence  of  an  interval 
—  8  •  •  •  +  8,  within  which  x(^=0)  lies,  and  such  that  the  upper 
limit  of  f(x}  y),  when  x  takes  a 
constant  value,  the  variable  y  being 
confined  to  the  interval  -\-  x  •  •  •  —  x, 
is  had  through  the  value  y  =  <£2(*)> 
and  the  lower  limit  througli 
y  =  ^  (x).  This  necessary  condition 
in  question  for  a  proper  maximum 
is  that  f(x,  4>%(x))  be  invariably 
less  than  f(0,  0),  and  for  a  proper  minimum  we  must  have 
invariably  f(x}  ^(x))  greater  than  f(0,  0). 

In  the  first  case  the  upper  and  lower  limits  of  f(x,  y)  are  both 
less  than  /(O,  0)  and  in  the  second  case  they  are  both  greater. 


-5 

F 

X         S 

y 
o 

y 

FIG.  10 


of 


Note  that  lim  $2  (x)  =  0,  since 


The  same  is  true 


The  same  conditions  must  be  true  with  regard  to  the  upper  and 
lower  limits  off(x,y)  with  constant  y  such  that  \y\<&,  the  vari 
able  x  being  limited  to  the  interval  —  y  •  •  •  -\-  y,  which  limits  are 
reached  through  the  values  ^2  (y)  and  /^r1  (y)  respectively. 

THEOREM  II.  The  fulfillment  of  all  the  conditions  made  above  is 
sufficient  that  f  (0,0)  be  a  proper  extreme  of  f(x,  y).  Accordingly 
f(0,  0)  is  a  proper  maximum  if  there  exists  a  positive  quantity  8 
such  that  we  have  simultaneously 


for     0< 


f(x,<t>i(x))<f(Qt  0), 


[1] 
and 

[2]  for     0<|y|<8,    /(^2(y),  y)  </(0,  0) 

with  corresponding  conditions  for  a  proper  minimum. 

To  prove  the  two  theorems  just  stated  we  remark  first  that  on 
account  of  the  continuity  of  f(x,  y)  with  respect  to  y  the  function 


FUNCTIONS  OF  TWO  VARIABLES  41 

f(x,  y}  with  constant  x  and  with  the  assumption  that  y  takes  all 
values  of  the  interval  —  x  -  •  •  -f  x  has  for  all  these  values  a  finite 
upper  limit  and  a  finite  lower  limit,  and  further  that  f(x,  y)  reaches 
these  limits  for  values  y  =  fa(x)  and  y  =  fa  (x)  (see  §  8). 
Hence  for  values  of  y  such  that 

[3]          |  y\  =§  |  x\  it  is  clear  that  f(x,  y)  ^f(x,  fa(x)). 

Furthermore,  in  virtue  of  the  definition  of  a  proper  maximum  of 
f(x,  y)  there  must  be  a  positive  quantity  8  such  that  if  only  \x\ 
and  \y\  are  smaller  than  8  we  must  have 

[4]  f(x,y)-f(0,0)<0. 

It  follows,  if  |  x  j  <  8  and  x  ¥=  0  and  if  we  substitute  y  =  fa  (x)  in 
[4]>that 


which  is  in  fact  the  inequality  [1]. 

Reciprocally  from  [1]  and  [3]  are  obtained  the  inequalities 

0<|*|<8     and    f(x,  y)  -  /"(O,  0)  <  0, 
where  |  y  |  =  |  x  |  <  8. 

If  the  relation  [4]  is  to  be  true  for  all  systems  of  values  (x,  y) 
where  \x  and  \y\  are  smaller  than  8  (excepting  x  =  0  and  y  =  0)? 
then  in  addition  to  [1]  we  must  have  the  corresponding  pair  of 
inequalities  [2],  which  may  be  derived  without  trouble. 

We  have  corresponding  conditions  for  improper  extremes  : 

THEOREM  III.  In  order  that  f(0,  0)  be  an  improper  maximum 
of  f(x,  y)  it  is  necessary  and  sufficient  that  there  exist  a  positive 
quantity  8  such  that  for  any  x  with  absolute  value  less  than  8 
the  value  f(x,  fa(x))  is  not  greater  than  f(0,  0)  and  for  any  y 
with  absolute  value  less  than  8  the  value  /(^^(y),  y)  is  not 
greater  than  f(0,  0  )  ;  while  at  the  same  time  corresponding  to 
every  positive  quantity  8f  which  is  less  than  8  there  is  at  least 
one  value  of  x  or  y  whose  absolute  value  is  less  than  8*  and  for 
which  either  f(x,  fa(x))  or  f(^(y),  y}  is  equal  to  f(0,  0),  The 
conditions  for  an  improper  minimum  follow  at  once. 


42  THEORY  OF  MAXIMA  AND  MINIMA 

THEOREM  IV.  That/(0,  0)  may  not  be  a  minimum  (proper  or 
improper)  of  f(x,  y)  it  is  necessary  and  sufficient  that  to  every 
positive  quantity  8  there  either  exists  a  quantity  xr,  with  absolute 
value  less  than  S,  such  that 

[5]  /(*',  <»,(*'))  </(»,  0), 

or  that  there  exist  a  quantity  y',  with  absolute  value  less  than  S, 
such  that 

[6]  /(^i(y').2/')</(0,  0); 

and  that  /(O,  0)  may  not  be  a  maximum  (proper  or  improper) 
°f  /(#»  2/)  it  is  necessary  and  sufficient  that  corresponding  to 
every  positive  quantity  &  there  may  be  found  either  a  quantity  x" , 
with  absolute  value  less  than  £,  such  that 

[7]  /(*",  $,(*"))  >/(0,0), 

or  a  quantity  y",  with  absolute  value  less  than  S,  such  that 

[8] 


CHAPTER  IV 

THE  SCHEEFFER  THEORY 

I.    GENERAL    CRITERIA    FOR    A    GREATEST    AND    A    LEAST 

VALUE   OF  A  FUNCTION   OF   TWO   VARIABLES;    IN    PARTIC 

ULAR  THE  EXTRAORDINARY  EXTREMES 

30.  The  theorems  of  Stolz  which  were  developed  in  the  pre 
ceding  article  are  closely  related  to  those  of  Scheeffer,  which  are 
of  more  practical  value  since  the  computations  required  have  to 
do  mostly  with  a  few  of  the  initial  terms  of  the  expansion  of 
f(x,  y)  —  /(O,  0)  in  ascending  positive  integral  powers  of  x  and  y. 
We  shall  assume  that  the  function  f(x,  y)  is  such  that  it  may 
be  expanded  by  the  Taylor-Lagrange  theorem  in  the  form 


=  /to  y)  +  W'x  to  +  M>y  +  ^)  +  ¥*  to  +  fa>y  +  0k)] 
=/to  y)  +  vito  y)+¥ito  y)  +  WfLx(x  +  Oh>y  +  M) 

+  2  hkfi',,  (x  +  0h,  y  +  9k)  +  &fyy  (x  +  0h,y  +  0k)],  etc., 
where  0<^<1. 

If  we  write  x=  0,  y  —  0  and  then  put  h  =  x,  h=y,  it  is  seen  that 
[1]  /to  y)-/(0,  0)  =  Gnto  y)4--Rn+1to  y), 

where  Gn(xt  y)  denotes  the  collectivity  of  terms  of  the  n  first 
dimensions  and  Rn+l(x,  y)  is  the  remainder  term  (Lagrange, 
Theorie  des  Fonctions,  Vol.  I,  p.  40). 

The  Scheeffer  theorem.  If  an  index  n  and  positive  quantities 
a  and  8  can  be  determined  to  satisfy  the  two  postulates  (1)  that  for 
all  values  of  x  such  that  0  <\x  <&  the  upper  and  lower  limits 
of  \GH(x,  y)\  =  a  x\n,  with  constant  values  of  x  and  with  y  limited 
to  the  interval  -x  ----  \-  x,  and  (2}  that  for  all  values  of  y  such 

43 


44 


THEORY  OF  MAXIMA  AND  MINIMA 


that  0<|y|<8  the  upper  and  lower  limits  of  \Gn(x,  y)\^a\y^, 
where  y  has  constant  values  and  where  x  lies  within  the  interval 
—  y  '  '  '  +  y>  then  the  two  functions  f(x,  y)  and  Gn  (x,  y)  have 
simultaneously  on  the  position  (0,  0)  either  a  proper  maximum  or 
a  proper  minimum. 

For,  let  the  lower  and  upper  limits  of  Gn  (x,  y},  with  constant  x 
and  with  \y\=  x\,  be  Gn(x,  ®i(x))  and  Gn(x,  ®2(x))  (see  §  29)5  an(i 
with  constant  y  and  with  x  \  ^  |  y  \  let  the  upper  and  lower  limits 
of  Gn(x,  y}  be  £n(¥2(y),  y)  and  ^(^(y),  y).  Since  Rn+l(x,  y} 
is  a  homogeneous  integral  function  of  the  (n  +  1)  th  dimension  in 
x,  y  and  consists  of  n  +  2  terms,  we  note  that  corresponding  to 
any  positive  quantity  e'  we  may  always  find  another  positive 
quantity  81  such  that  if 


\y\^\x\  and  0<| 
and  also  such  that  if 
x^        and  0< 


',  then 


',  then  \Rn+1(x,  y)\  <  (n  + 


Hence  writing  (n  -f  2)  e'  x  =  e  and  (n  +  2)  e'  |  y  |  =  e,  and  denoting 
the  corresponding  value  of  &  by  8,  it  is  seen  that  there  is  always 
an  interval  —  S  •  •  -  +  8  such  that  if 


[2]      0<\x 
and  if 

[3]       0<|^|<Sand 


and  \y\  =i  \x\,  then 


<e 


€  |y|,  then 


,  y)\  <e|y|- 


It  follows  then  from  [1]  and  [2]  that  for  values  of  x,  y  such 
that  \x\  <  8  and  \y\  ^  a;|  we  have 

[4]  Gn(xt3>l(x))-e\x\»<f(x,y)-f(Q,  0) 


and  from  [1]  and  [3]  that  for  values  of  xy  y  such  that  0  <  \y\  <  8 


and 


^  I  y  I  we  have 


[5] 


-/(0,  0) 


THE  SCHEEFFER  THEORY  45 

If  next  we  assume  that  £n(0,  0)  is  a  proper  extreme  of 
Gn(x,y)  and  that  the  two  postulates  of  the  theorem  have  been 
satisfied,  then  if  6rn(0,  0)  is  a  minimum  it  is  evident  for  small 
values  of  x  and  y  that  Gn(x,  ^(x))  and  Gn(^1(y),y)  are  positive 
quantities,  and  from  the  postulates  it  follows  that  for  values 

0<  x\<8  and  |y|^  x  we  have  Gn(xy  4>1(^))^  a\x\n 
and  for  values 

0  <  \y  |  <  8  and  \x\  ^  \y  \  we  have  Gn(Vl  (y),  y)  ^  a  y  \ 

Accordingly  it  follows  from  [4]  for  values 
[6]  0<|z|<Sand  \y\^\x\  that  (a  -  e)\x\»<f(x,  y)  -/(O,  0); 
and  from  [5]  for  values 

[7]   0<|*/|<Sand  \x  =i  \y\  that  (a  -  e)\y\»<f(x,  y)-/(0,  0). 

Since  e  may  be  made  smaller  than  «,  it  follows  in  both  [6]  and 
[7]  that/(#,  y)—  /(O,  0)  is  positive  and  consequently  that/(0,  0) 
is  a  proper  minimum  of  f(x,  y)  (see  Stolz's  second  theorem,  §  29). 

If  6rw(0,  0)  is  a  proper  maximum  of  Gn(xt  y),  then  with  small 
values  of  x  and  y  the  expressions  Gn(x,  ®z(x))  and  Gn(V%(y)t  y} 
must  be  negative. 

Hence,  due  to  the  postulates  for  values 

0<|^|<3and  |y|^  x\,  we  have  Gn(x,  4>2(a;))^_  a\x\*, 
and  for  values 


and  in  a  similar  manner  as  above  it  follows  that  /(O,  0)  is  a  proper 
maximum  of  f(x,  y). 

31.  Stolz's*  added  theorem.  If  Gn(0,  0)  is  not  an  extreme  of 
Gn  (x,  y),  the  following  conditions  are  sufficient  to  make  it  impossible 
that  f(0,  0)  should  le  an  extreme  of  f(x,y):  if  (1)  for  all  positive 
values  of  x  and  y  such  that  0  <  \  x  \  <  8  and  0  <  \  y  \  <  8,  or  for  all 
negative  values  within  the  same  limits,  at  least  one  of  the  two  upper 
limits  of  Gn(x,  y)  defined  above  is  positive  and  not  less  than  a\x\n  or 

*  Stolz,  p.  218, 


46  THEORY  OF  MAXIMA  AND  MINIMA 

a\y\n  respectively,  and  (2)  for  all  positive  values  of  x  and  y  such 
that  0  <  x  |  <  £  and  0  <  |  y  \  <  3,  or  /or  all  negative  values  within 
the  same  limits,  at  least  one  of  the  two  lower  limits  of  Gn(x,  y) 
defined  above  is  negative  and  not  greater  than  —a  x\n  or  —  a\y\n 
respectively;  that  is,  if,  under  the  restrictions  just  made,  Gn(x,  <j>2(x)) 
is  positive  and  Gn(x,  &i(x))  negative,  or  if  Gn  (SPg  (y),  y)  is  positive 
and  G^^y},  y)  negative. 

If  we  limit  x,  for  example,  to  the  interval  0  ...  S,  and  if  we 
suppose  that  the  following  inequalities  Gn  (x,  <l>2  (x))  ^  a  x  n  and 
Gn  (x,  Ql  (x))  ^  —  a  x  n  exist,  it  is  seen  that  these  two  expressions 
vanish  only  for  x  =  0. 

From  [1]  and  [2]  it  follows  for  y  =  Q^x)  and  y=  <&2(x) 
for  values  of  x  within  the  interval  in  question 


f(x,  ^(^ 
and  f(x,  <S>a(a))-/(0,  0)  >  (a  -  e}\x\\ 


Since  we  may  take  e<a,  it  is  seen  that  in  the  two  expressions 
just  written,  the  difference  on  the  left-hand  side  is  in  the  first  case 
negative  and  in  the  second  case  positive,  so  that  /(O,  0)  is  not  an 
extreme  of  f(xt  y}  (see  Stolz's  fourth  theorem,  §  29). 

32.  The  analytic  proof  given  in  §  30  of  the  Scheeffer  theorem 
is  essentially  due  to  Stolz.  Owing  to  its  importance  we  shall  give 
Scheeffer's  statement  of  this  theorem  with  his  geometric  deductions 
(Math.  Ann.,  Vol.  XXXV,  p.  553). 

The  Scheeffer  theorem  otherwise  stated.  Let  f(x,  y)  be  any 
function  as  already  defined  of  x,  y  which  vanishes  at  the  origin* 
and  let  its  behavior  in  the  neighborhood  of  this  point  be  suffi 
ciently  explicit  for  the  determination  regarding  the  appearance 
of  extremes  by  means  of  power  series  to  be  possible  ;  in  other  words, 
ive  assume  that  there  exists  a  power  arn  such  that  upon  every 
circle  described  about  the  origin  as  center,  whose  radius  r  is  not 
smaller  than  a  definite  quantity  g,  the  greatest  and  the  least  values 
of  the  function  f(x,  y},  viz.,  fi(r)  and  /2(r)  for  all  points  of  the 

*  If  /(O,  0)  ^  0,  we  must  write  /(x,  y)  —  /(O,  0)  in  the  place  of  f(x,  y)  in  the  present 
discussion. 


THE  SCHEEFFER   THEORY  47 

circumference  of  the  circle  with  radius  r,  are  in  absolute  value 
greater  than  arn.  Then  in  the  Taylor-Lagrange  development 

given   above  ,.  „    .       .      „       ,       x 

f(xt  y)  =  Grn  (x,  y)  +  Mn  + 1  (x,  y), 

where  R  +  1(#,  y)  consists  of  all  terms  beyond  those  of  the  nth 
dimension,  the  integral  rational  function  Gn  (x,  y)  behaves  in  the 
neighborhood  of  the  origin  as  does  the  function  f(x,  y).  For,  as 
we  shall  show,  in  the  first  place  the  greatest  and  the  least  values 
of  both  functions  correspond  with  respect  to  sign  for  every  small 
radius  r,  and  from  this  it  follows  that  there  appear  simultan 
eously  at  the  origin  extremes  for  both  functions,  if  such  extremes 
exist ;  and  secondly,  if  a'  is  any  quantity  situated  between  0  and  a, 
then  upon  the  circumference  of  every  circle  with  radius  r  (within 
a  certain  limit  g')  the  greatest  and  the  least  values  of  the  function 
Gn(x,  y)  are  in  absolute  value  greater  than  arrn,  and  from  this 
it  follows  also  that  the  degree  of  distinctness  that  marks  the 
behavior  of  Gn(x,  y)  is  the  same  as  that  of  f(x,  y}. 

It  is  evident  that  we  may  replace  x  and  y  in  the  remainder 
term  Rn  +  i(%,  y}  by  r,  where  r-is  the  radius  of  the  small  circle 
about  the  origin  within  which  the  point  (x,  y)  is  situated ;  and 
at  the  same  time  we  may  replace  all  coefficients  by  their  absolute 
values.  In  this  way  we  have  for  the  absolute  value  of  Rn  +  l(x,  y) 

an  upper  limit  Arn+\  We  shall  take  the  radius  r  smaller  than  -- 
so  that  arn  >  Arn  +  \ 

Since  f^r)  and  fz(r)  are  by  hypotheses  greater  in  absolute 
value  than  arn,  it  follows  from  the  equation 


that  those  values  of  x,  y  on  the  periphery  of  the  circle  with 
radius  r  which  give  f^r)  and  f2(r),  cause  Gn(x,  y)  and  fn(x,  y} 
to  have  the  same  sign.  If/^r)  and/2(r)  have  the  same  sign,  it 
follows  from  the  above  expression  that  the  greatest  and  least 
values  of  Gn(x,  y}  have  this  same  sign.  If  the  two  quantities 
/!  (r)  and  /2  (r)  have  contrary  signs,  the  same  is  true  of  Gn  (x,  y) 
for  those  values  of  x,  y  which  produce  /x  (r)  and  /2  (r) ;  and 


48  THEOKY  OF  MAXIMA  AND  MINIMA 

consequently  for  a  greater  reason  the  greatest  and  least  values 
of  Grn(x,  y)  have  contrary  signs. 

The  second  part  of  the  theorem  follows  in  the  same  way  if 

we  take  the  radius  r  not  only  smaller  than  —  but  also  so  small 

A  f 

that  arn  —  Arn  +  l>a'rn;  that  is,  if  we  put  g1  equal  to and 

A 

take  r  less  than  g'.  It  is  then  evident  that  the  values  of  x,  y 
which  produce  fi(r)  and  /2(r)  when  written  in  the  expression 

G*(x>  y}  =f(x>  y}  -  fin+i  (^  y} 

cause  the  right-hand  side  to  be  in  absolute  value  greater  than  a'rn 
when  /!  (r)  and  /2  (r)  have  the  same  sign ;  and  when  these  two 
quantities  have  contrary  signs  the  corresponding  values  of 
Gn(x,  y)  wiU  m  absolute  value  be  greater  than  a'rn,  and  the 
same  must  a  fortiori  be  true  of  the  greatest  and  the  least 
values  of  Gn(x,  y}. 

33.  If,  however,  we  cannot  find  an  integer  n  and  a  quantity  a' 
which  satisfy  the  conditions  above,  we  can  make  no  conclusions 
regarding  the  behavior  of  the  function  f(x,  y)  by  means  of  powers 
series  and  by  using  the  method  indicated.  For  in  this  case  we 
shall  show  by  means  of  simple  examples  which  follow  this  chap 
ter  that  in  some  cases  the  function  Gn(x,  y}  is  invariably  positive, 
while  f(xt  y)  may  be  also  negative ;  and  in  some  cases  Gn(x,  y) 
may  be  both  positive  and  negative  while  f(xt  y)  retains  a  con 
stant  sign  (see  Ex.  3,  p.  61,  and  Prob.  2,  p.  62).  But  if  the 
conditions  of  Scheeffer's  theorem  exist  it  is  seen  that  the  in 
vestigation  of  the  function  f(x,  y)  has  been  reduced  to  that  of 
the  function  Gn(xt  y)\  in  other  words,  the  investigation  has 
resolved  itself  into  the  question :  How  can  we  recognize  whether 
a  limit  g'  and  a  quantity  a'  exist  such  that  ^lpon  every  circle 
with  radius  r<g'  the  greatest  and  the  least  values  of  a  given 
integral  function  of  the  nth  degree  Gn  (x,  y)  are  in  absolute  value 
greater  than  a'rn  ?  And  how  can  we  eventually  fix  the  signs  of 
these  greatest  and  least  values  and  thereby  determine  the  extremes 
of  the  function  Gn  (x,  y)  ? 

These  questions  we  shall  now  answer. 


THE  SCHEEFFER  THEORY  49 

II.   HOMOGENEOUS  FUNCTIONS 

34.  In  the  expansion  of  f(x,  y)  —  /(O,  0)  suppose  that  the  first 
terms  that  appear  form  a  homogeneous  function  of  the  7ith  degree 
in  x  and  y  which  is  the  function  Gn(x,  y).  With  respect  to  such  a 
function  there  are  three  cases  to  consider,  according  as  this  func 
tion  is  a  definite  form,  an  indefinite  form,  or  a  semi-definite  form 
(see  §  13).  If  we  write 


it  is  seen  that  Gn(x,  y)  changes  upon  every  straight  line  through 
the  origin  proportionally  to  the  nth  power  of  r.  If  then  G1  and 
G2  are  the  greatest  and  the  least  values  of  Gn(x,  y)  upon  the 
periphery  of  the  unit  circle,  then  G^rn  and  G2rn  are  the  greatest 
and  least  values  upon  any  arbitrary  circle  r. 

The  signs  of  G1  and  G2  may  be  obtained  directly  through 
decomposing  Gn  (x,  y)  into  its  linear  factors,  which  may  be  found 
by  solving  an  equation  of  the  ?^th  degree.  For  we  may  write 

G 


(\ 
1 ,  ^  \  =  xng  (u) ,     where     -  =  u 
X/  X 

and  g(iC)  is  an  integral  function  of  the  ?tth  degree  hi  u.  Owing  to 
the  fundamental  theorem  of  algebra,  g(u}  may  be  decomposed 
into  factors  which  are  linear  and  quadratic  with  negative  dis 
criminants  if  we  restrict  all  the  coefficients  to  real  quantities;  or 
these  factors  are  all  linear  if  we  allow  imaginary  coefficients,  the 
quadratic  factors  breaking  up  into  two  imaginary  linear  com 
ponents.  If  these  factors  are  multiplied  by  the  respective  powers 
of  x,  we  have  the  corresponding  decomposition  of  Gn(x,  y)  into 
its  linear  and  quadratic  factors.  At  the  outset  it  is  clear  that 
if  the  degree  of  Gn(x,  y)  is  odd,  then  G±  and  G2  must  be  equal 
but  of  opposite  sign,  since  Gn(x,  y)  changes  sign  when  x,  y  are 
changed  into  —  x,  —  y.  Furthermore,  note  that  Gn(a)x,  (oy)  = 
<onGn(x,  y),  where  o>  is  a  positive  quantity.  It  follows  that  if 
Gn(x,  y}  is  positive,  negative,  or  zero,  then  Gn(o>x,  o>y)  is  positive, 
negative,  or  zero. 


50  THEORY  OF  MAXIMA  AND  MINIMA 

If  Gn(x,  y)  is  an  indefinite  form,  there  are  values  x,  y  which 
give  Gn(x,  y)  a  positive  value,  and  other  values  x,  y  which  give 
it  a  negative  value.  Let  §  be  a  positive  quantity  however  small. 
It  is  seen  that  by  a  proper  choice  of  co  we  may  find  values  of  xt 
y  where  \x  <  8  and  \y\  <  8  such  that  Gn(x,  y}  is  positive,  and  other 
systems  of  values  x,  y  within  the  same  interval  for  which  Gn  (x,  y) 
is  negative.  Accordingly  the  value  Gn(Q,  0)  is  not  an  extreme  of 

0,(*|f):* 

If,  however,  n  is  even,  and,  first,  if  the  linear  factors  of  Gn(x,  y) 

are  all  imaginary,  then  Gn(x,  y)  cannot  change  sign  nor  vanish.  It 
is  a  definite  form  and  the  quantities  Gl  and  G2  have  the  same 
sign.  If,  secondly,  there  are  real  linear  factors,  and  if  at  least  one 
enters  to  an  odd  degree,  then  Gn(x,  y)  takes  both  signs.  Gn(x,  y) 
is  then  an  indefinite  form  and  the  sign  of  G±  is  different  from 
that  of  6r2.  It  thirdly,  there  enter  real  linear  factors,  but  each 
only  to  an  even  degree,  the  form  Gn(xt  y}  may  vanish  but  it 
cannot  change  sign.  It  is  a  semi-definite  form,  and  one  of  the 
extremes  Gl  and  G2  is  zero.  In  this  case  by  a  proper  choice  of 
co  above  it  is  seen  that  Gn(x,  y}  vanishes  for  values  of  x,  y  other 
than  zero  and  situated  within  the  interval  |  x  \  <  B  and  j  y  \  <  8. 
In  this  case  Gn(Q,  0)  is  an  improper  extreme  of  Gn(x,  y)\  and 
the  behavior  of  f(xt  y)  at  the  origin  cannot  be  recognized  with 
out  further  discussion. 

In  all  cases  t  except  the  last  a  positive  quantity  a'  may  be  so 
determined  that  upon  every  arbitrary  circle  r  the  greatest  and 
least  values  of  the  function  Gn(xt  y),  viz.,  G-^r11  and  G%rn,  are  in 
absolute  value  greater  than  a'r11 ;  for  we  need  only  take  ar  smaller 
than  the  absolute  values  of  6^  and  6r2.  In  these  cases  (again 
excepting  the  last)  there  are  found  in  a  sufficiently  distinct 
manner  (in  the  previous  precise  sense  of  the  word,  see  §  27)  either 
a  maximum  or  a  minimum  of  the  function  G(x,  y),  or  there  does 
not  exist  such  an  extreme. 

The  decomposition  of  Gn(x,  y)  into  its  linear  factors  is  not 
necessary,  since  we  may  determine  the  sign  of  G^  and  G%  by 

*  Cf.  Stolz,  p.  222. 

t  The  discussion  is  for  the  most  part  due  to  Scheeffer,  loc.  cit. 


THE  SCHEEFFEK  THEORY  51 

means  of  elementary  algebraic  operations.    For  we  may  determine 
the  multiple  factors  of  Gn  (x,  y)  and  write  this  function  in  the  form 


where,  in  general,  ^rk  is  an  irreducible  factor  of  the  &th  degree  in 
x  and  y  with  integral  coefficients  and  \k  denotes  the  number  of 
times  this  factor  occurs.  Then  by  Sturm's  theorem  we  may  deter 
mine  for  each  such  function  tyk(z,  y)  the  number  of  real  factors 
and  by  \k  the  number  of  times  such  factor  is  repeated. 

The  theory  just  outlined  of  the  integral  homogeneous  functions 
offers,  owing  to  the  Scheeffer  theorem  for  the  general  theory  of 
maxima  and  minima  of  arbitrary  functions,  the  following  theorem  : 

If  in  the  development  of  the  function  f(x,  y)  in  powers  of  x,  y 
all  terms  of  the  first  to  the  (n  —  V)th  dimensions  are  identically  zero, 
while  the  terms  of  the  nth  dimension  constitute  a  form  Gn(x,  y) 
homogeneous  in  x  and  y,  and  if,  first,  Gn  (x,  y)  is  an  indefinite 
form  (which  is  always  the  case  if  n  is  odd),  then  on  the  position 
(0,  0)  there  is  neither  a  maximum  nor  a  minimum  of  the  function 
f(x,  y)  ;  if,  secondly,  Gn  (x,  y)  is  a  definite  form,  there  enters  accord 
ing  to  the  sign  of  this  form  an  extreme  of  f(x,  y);  if,  finally, 
Gn  (x,  y)  is  semi-definite,  the  behavior  of  the  function  f(xt  y)  cannot 
be  recognized  from  the  behavior  of  Gn  (x,  y}. 

From  this  theorem  it  follows  that  if  /(O,  0)  is  an  extreme  of 
f(x,  y},  the  terms  of  the  first  dimension  of  the  expansion  by 
Taylor's  formula  of  f(x,  y}—  /(O,  0)  must  be  wanting,  and  conse 
quently  we  must  have 

fi  (0,  0)=0     and    .f,;(0,  0)=  0. 
If,  furthermore, 

0)  =  Ax*  +  2  Bxy  +  O/2+ 


then  /(O,  0)  is  not  or  is  (in  fact  a  proper)  extreme  of  f(x,  y) 
according  as  A  C  —  B2  is  negative  or  positive.  If  this  discriminant 
is  positive,  then  /(O,  0)  is  a  maximum  or  a  minimum  according 
as  A  and  C  (which  necessarily  have  one  and  the  same  sign)  are 
negative  or  positive  (see  §  14). 


52  THEORY  OF  MAXIMA  AND  MINIMA 

But  if  AC  —  B2=  0,  a  criterion  regarding  an  extreme  of  f(x,  y) 
with  the  help  only  of  the  terms  of  the  second  dimension  cannot  be 
had.  We  must  then  take  in  addition  terms  of  the  third,  fourth,  •  •  • 
degrees  in  the  above  expansion  of  f(x,  y)  in  order,  if  possible,  to 
satisfy  the  postulates  of  Scheeffer  regarding  the  function  Gn(x,  y), 
In  this  case  we  may  write,  if  A  is  different  from  zero, 


where  ^,  JJ,  •  •  •  denote  the  collectivity  of  the  terms  respectively 
of  the  third,  fourth,  •  •  •  dimensions  in  x,  y. 

If  in  this  expression  we  write  x  =  Bt,  y  =  —  At,  it  is  seen  that 

f(Bt,  -At)- 


and  if  the  constant  A3  is  different  from  zero,  it  is  seen  that  by 
giving  positive  and  negative  values  to  t,  the  above  expression 
may  take  both  positive  and  negative  values,  so  that  there  is  no 
extreme  of  f(x,  y)  on  the  position  (0,  0). 

But  even  if  the  first  term  that  appears  on  the  right  of  the 
expansion  in  t  is  of  even  degree,  we  cannot  conclude  that  there 
is  an  extreme,  as  is  illustrated  by  the  classic  example  of  Peano 
(see  §  24),  viz.,  f(x,  y}  =  Ayz+2  Bx*y  +  Cx*. 

Further  investigation  is  therefore  necessary  when  the  terms  of 
the  second  degree  constitute  a  semi-definite  form,  and  this  case 
is  continued  in  the  following  sections. 

III.    EXTREMES  OF  THE  FUNCTION   Gn(x,  ?/),  INTEGRAL 
IN  x  AND  y,  WHICH  IS  NOT  HOMOGENEOUS 

35.  We  must  next  determine  whether  or  not  the  value  Gn(Q,  0) 
is  an  extreme  of  Gn(x,  y)  when  this  function  is  not  homogeneous 
in  x  and  y  and  when  the  terms  of  the  lowest  dimension  in  Gn(x,  y) 
constitute  a  semi-definite  form.  We  must  again  raise  the  question 
regarding  the  existence  of  an  expression  a'rn  which  for  all  suffi 
ciently  small  values  of  r  is  to  be  smaller  than  the  absolute  values 
of  the  greatest  value  and  of  the  smallest  value  of  Gn(x,  y)  upon 
the  periphery  of  a  circle  of  radius  r,  where  r  is  sufficiently  small. 


THE  SCHEEFFER  THEORY  53 

In  order,  then,  to  acquaint  ourselves  with  the  .different  possi 
bilities  which  may  enter  in  the  behavior  of  the  function  Gn(xt  y) 
at  the  point  (0,  0),  we  take  a  small  circle  with  radius  r  and  seek 
upon  it  the  two  positions  at  which  the  function  Gn(x,  y)  takes 
its  greatest  and  its  least  value.  Call  these  values  the  extreme 
values  of  Gn(x,  y).  They  are  found  (see  §  15)  by  solving  the 
three  equations  ^ 


By  eliminating  X  from  the  first  two  of  these  equations  we  have 
an  equation  of  the  ?ith  degree 

y^-*^  =  0,  (*) 

dx  dy 

an  equation  which  is  satisfied  by  all  values  of  x  and  y  which 
offer  extreme  values  of  Gn(x,  y}  upon  any  arbitrary  circle  r. 

It  is  known  in  the  theory  of  algebraic  functions  that  every 
branch  of  an  algebraic  curve  of  the  ?zth  order  which  contains  the 
origin  may  be  expressed  in  the  neighborhood  of  the  origin  through 
an  independent  variable  (£,  say)  in  the  form 

x  =  ak  4-  ak2  -f 


and  this  expression  for  the  curve  may  be  made  in  any  number  of 
different  ways  such  that  in  each  of  the  series  for  x  and  y  the  first 
coefficient  which  is  different  from  zero  (in  case  there  is  one)  has  an 
exponent  which  is  ^  n.  It  follows  that  both  those  branches  which 
include  the  origin  of  the  curve  (*),  and  whose  points  of  intersection 
with  the  circles  of  small  radii  offer  the  extreme  values  of  Gn(xt  y) 
upon  these  circles,  may  be  expressed  in  the  form  (ii)  through  an 
independent  parameter  kt  so  long,  at  least,  as  we  remain  in  the 
immediate  vicinity  of  the  origin  ;  that  is,  so  long  as  very  small 
values  are  ascribed  to  k.  We  shall  call  these  two  branches  the 
two  extreme  curves  of  the  function  Gn(x,  y). 


54  THEORY  OF  MAXIMA  AND  MINIMA 

36.  We  must  next  distinguish  between  the  cases  (1)  when 
(excepting  for  isolated  values  of  r)  the  extreme  values  of  Gn(x,  y) 
are  both  different  from  zero  and  (2)  when  one  of  these  extremes 
is  zero. 

If  both  extremes  are  different  from  zero,  then  the  expression 
Gn(x,  y),  if  we  write  for  x  and  y  the  two  series  (ii)  which  corre 
spond  to  an  extreme  curve,  will  begin  with  a  term  Akm,  which  for 
small  values  of  k  determines  both  the  sign  and  the  order  of  magni 
tude  of  the  entire  expression.  This  order  is  the  mth  order  if  we 

7?7 

consider  k  a  quantity  of  the  first  order,  and  it  is  of  the  —  th  order  if 

P 

we  consider  &M  the  first  order,  where  If  is  the  smallest  exponent 
that  actually  appears  in  (ii).  The  number  //.,  as  we  saw  above,  can 
at  most  be  equal  to  n.  We  have  similar  quantities  A',  mr,  ///  for 
the  second  extreme  curve.  If  the  two  numbers  m  and  m'  are  not 
both  even,  there  can  be  no  maximum  nor  minimum  of  GH(x,  y)  at 
the  origin,  since  this  function  in  this  case  changes  sign  with  k 
upon  an  extreme  curve.  The  same  is  true  if  m  and  m1  are  even 
numbers  while  A  and  A'  have  opposite  signs,  for  then  the  function 
Gn(x,  y)  shows  different  signs  upon  the  two  extreme  curves. 

If,  finally,  m  and  m'  are  both  even  while  A  and  Af  have  the  same 
sign,  then  we  have  a  maximum  or  minimum  of  Gn  (x,  y)  according 
as  this  sign  is  negative  or  positive. 

In  all  three  cases  it  is  clear  that  a  quantity  a'  and  an  upper 
limit  g'  of  r  may  be  so  determined  that  for  r<g'  the  values  of 
Gn(x,  y)  upon  both  extreme  curves  are  everywhere  in  absolute 
value  greater  than  afrp,  where  p  is  the  greater  of  the  two 

,          m        ,    m' 
numbers    —   and   — -  • 

/*  F 

If,  however,  the  value  of  Gn  (x,  y}  is  invariably  zero  upon  one  of 
the  extreme  curves,  there  cannot  be  a  maximum  or  minimum  at 
the  origin,  nor  is  there  an  expression  a'rp  of  the  kind  required 
above.  But  this  can  only  occur  when  Gn(x,  y)  contains  a  squared 
factor  which  when  put  equal  to  0  defines  a  real  double  curve 
that  passes  through  the  origin ;  for  otherwise,  with  the  vanish 
ing  of  Gn  (x,  y)  upon  crossing  the  circumference  of  any  circle  with 


THE  SCHEEFFER  THEORY  55 

radius  r,  there  must  be  a  change  of  sign  in  Gn(x,  y).  The  squared 
factor  enters  as  a  factor  to  the  first  power  in  (ii),  so  that  points 
on  this  curve  make  Gn(xt  y)  identically  zero. 

In  the  sequel  we  shall  assume  that  such  factors  have  been  di 
vided  out  of  Gn(x,  y),  so  that  the  case  in  question  does  not  enter. 

Under  this  assumption,  which  must  be  tested  in  every  indi 
vidual  case,  there  exists,  in  virtue  of  the  considerations  already 
laid  down,  always  a  smallest  number  p  associated  with  which  a 
constant  a'  and  an  upper  limit  g'  of  the  radius  r  may  be  so  de 
termined  that  upon  every  circle  of  radius  r<gf  the  two  extreme 
values  of  Gn(x,  y}  are  in  absolute  value  greater  than  a'rp;  and,  in 
fact,  this  number  p  (if  the  order  of  r  is  taken  as  unity)  expresses 
the  degree  of  the  magnitude  of  the  function  Gn  (x,  y}  upon  that  one 
of  the  two  extreme  curves  upon  which  this  order  is  the  highest. 
If  p  is  at  most  equal  to  ?i,  then  a'rp  for  small  values  of  r  is  not 
smaller  than  a'r11,  and  the  two  extreme  values  of  Gn  (x,  y)  are  there 
fore  certainly  greater  in  absolute  value  than  a'rn  ;  but  if  p  is  greater 
than  n,  then  for  small  values  of  r  at  least  one  of  the  extreme 
values  of  Gn  (xt  y)  is  in  absolute  value  smaller  than  arn,  however 
the  constant  a  may  be  chosen. 

It  is  thus  seen  that  in  virtue  of  the  fundamental  theorem  the 
function  Gn(xy  y}  may  be  used  as  a  criterion  for  determining  the 
existence  of  a  maximum  or  minimum  of  the  function  f(x,  y), 
where  Gn  (x,  y)  consists  of  the  terms  of  the  first  to  the  nth  order 
of  f(x,  y]  only  when  the  characteristic  exponent  p  is  at  most 
equal  to  n. 

37.  If  in  an  example  we  wished  to  discuss  the  function  Gn(x,  y} 
in  the  manner  indicated  above,  we  must  calculate  the  coefficients 
of  (ii),  which,  in  general,  is  a  somewhat  complicated  operation. 
The  following  method  leads,  however,  indirectly  to  the  same 
result,  viz.,  that  of  finding  the  extreme  values  of  Gn  (x,  y},  and  thus 
offers  an  easy  method  for  the  criteria  in  question.  The  method 
in  question  is  first  to  make  use  of  the  Stolzian  theorems  of  §  29, 
and  then  by  applying  the  Scheefferian  theorem  we  may  reach  the 
desired  conclusions.  Accordingly  we  must  determine  the  upper 
and  lower  limits  of  Gn(x,  y)  with  constant  x  and  |y|^|*|  as  well 


56  THEORY  OF  MAXIMA  AND  MINIMA 

as  the  upper  and  lower  limits  of  this  function  with  constant  y 
and  | a; |  =  \y\.  For  brevity  put  G  =  Gn(x,  y). 

The  values  of  y,  viz.,  y  =  ^^(x)  and  y^&^x),  which  offer 
the  first-mentioned  pair  of  limits,  fall  either  within  the  interval 
—  x  •  • '  •  +  x  or  upon  one  of  the  end-values  y  =  —  x  or  y  =  +  x. 
When  they  fall  within  the  interval,  since  Gn(x,  y)  is  a  continuous 
function  which  has  a  first  derivative  with  respect  to  y,  it  is  seen 

that  y  =  3>1  (x)  and  y  =  4>2(^)  are  solutions  of  the  equation  — -  =  0. 

dy 

In  the  second  case,  when  they  fall  upon  the  end-points  of  the  inter 
val,  then  y  =  x  OT  y  —  —  x  may  offer  the  desired  limit  or  limits. 

It  is  permissible  throughout  the  whole  discussion  to  fix  a  posi 
tive  quantity  a<\  as  the  upper  limit  for  \x\,  where  a  is  taken  so 
small  that  y  =  <E>2  (x)  and  y  =  4>x  (x)  are  convergent  series  in  x, 

which  when  substituted  in  the  equation  —  =  0  identically  satisfy 

dy 

it.   Furthermore  (see  §  29),  since  lim  <J>j  (x)  and  lim  4>2  (x)  =  0,  it  is 

x=0  x  =  0 

seen  that  no  constant  term  can  enter  these  expressions. 

The  method  of  determining  the  different  values  of  y  which 

f)C* 
satisfy  the  equation  —  =  0  is  found  in  §§  139  et  seq.    Let  these 

values  be 

P^x),     P2(x)t     P3(x)3.  (i) 

38.  We  may  next  see  which  of  these  functions  may  be  neglected 
from  the  investigation.  If  P(x)  denotes  any  of  the  functions 
PI(X) (i  =  1,  2,  •  .  •)  and  if  P(x)  has  the  form 

(1)  P(x)  =  xi>{a  +  x*R(x)},    where    p  >  0     and    <r>0, 

then  to  any  arbitrarily  chosen  e  >  0  there  corresponds  a  quantity 
S  >  0  such  that  there  are  values  x  \  <  8  for  which  |  x*R  (x)  |  <  e ; 
and  for  such  values  of  x  we  have 

(2)  \P(x)\>\0\{\a-e}. 

If  p  lies  within  the  interval  0</o<l  and  if  \x\  is  further  so 
diminished  that  \a  —  e>|^|1~p,  then  from  (2)  it  is  seen  that 
|P(#)|>  x\'  and  consequently  y  =  P(x)  would  fall  without  the 
fixed  interval  —  x  -  •  •  +  a;  We  see,  therefore,  that  any  series  which 


THE  SCHEEFFER  THEORY  57 

begins  with  a  term  ax*  +  -  -  >,  where  0  <  p  <  1,  may  be  neglected 
from  the  number  of  functions  given  in  (i). 

If,  next,  p  =  1  and  a  >  1,  we  may  take  e  so  small  in  (2)  that 
a|—e>l,  and  consequently  |P(«)|>|«|,  so  that  such  series  may 
also  be  neglected. 

Furthermore,  if  one  of  the  series  (i)  begins  with  +1  •  x  or  —  1  •  .t, 
and  if  the  second  term  has  the  same  sign  as  the  first,  then  evi 
dently  |P(a?)|>|«|,  and  such  a  series  may  accordingly  be  neglected 
from  the  investigation. 

39.  The  remaining  series  in  (i),  together  with  the  values  which 
correspond  to  the  end-points,  viz.,  y  =  +  x  and  y  =  —  x,  give, 
when  substituted  in  G  (x,  y},  the  following  functions : 

G(x,  -  x)t   G(x,  +  x),   G(x,  P,(x))}   G(x,  P2(x)),    -  .  - ;  (ii) 

and  we  have  to  determine  which  of  these  functions  presents  the 
upper  and  the  lower  limits  of  the  function  G  (x,  y)  for  the  interval 
in  question. 

By  taking  a(<l)  sufficiently  small  the  first  term  in  any  of 
the  functions  (ii)  is  as  a  rule  sufficient  in  determining  which 
will  give  the  required  upper  and  lower  limits.  Of  course,  if  two 
of  the  functions  (ii)  have  their  initial  terms  the  same,  it  may  be 
necessary  to  introduce  their  second  and  higher  terms  to  determine 
which  furnish  the  required  limits. 

Of  those  functions  whose  first  terms  are  negative  the  one  with 
smallest  exponent  gives  the  lowest  limit;  and  if  two  series  have 
the  same  negative  exponent,  the  one  with  greater  coefficient 
offers  the  lower  limit.  If  there  is  no  function  in  (ii)  whose  first 
term  is  negative,  then  in  determining  G(x,  &i(z))  we  note  that 
of  those  functions  whose  first  terms  are  positive  that  one  with 
highest  exponent  offers  the  lowest  limit ;  while  if  two  functions 
have  first  terms  with  the  same  exponent,  the  one  with  smaller 
coefficient  offers  the  lower  limit.  These  observations  must  be 
made  with  both  positive  and  negative  values  of  x,  where  |  x  <  a. 
If  one  of  the  functions  in  the  series  (ii)  is  zero,  while  the  others 
all  begin  with  a  positive  term,  then  G(x,  ®l(z))=Q,  etc.  We 
proceed  in  the  same  way  in  determining  G(xt 


58  THEORY  OF  MAXIMA  AND  MINIMA 


40.  To  determine  G(^r1(y)>y)  and  6r  (^2  (y),  y),  taking  y  con 
stant,  we  limit  x  to  the  interval  —  y  •  •  •  +  y.    Denote  by 


those  values  of  x  which  expressed  in  power  series  in  terms  of  y 

satisfy  the  equation  —  =0. 

ex 

The  two  limits  in  question  are  to  be  found  among  the  functions 


the  method  of  procedure  being  the  same  as  above.        . 

When  each  of  the  four  limits  G  (^1  (x),  x),  etc.  has  been  deter 
mined  for  values  of  x  within  the  fixed  intervals,  the  Stolzian 
theorem  is  at  once  applicable.  If  the  expansion,  say,  of  G(xt  &i(x)) 
is  akxk  +  ak+1xk  +  1+  -  -  -  and  if  k^n,  we  may  at  once  find  a 
constant  e  such  that 


and  if  the  same  is  true  of  the  three  other  limits  the  Scheefferian 
theorem  is  at  once  applicable. 

41.   Exceptional  cases.    If  the  f  unction  G  (x,  y)  contains  factors, 
say  x  ±y,  then   G  (x,  -P  x)  identically  vanishes.    More  generally 

O/~f 

the   equations   G(x,y)=Q  and  —  =  0   may  be  satisfied  by  the 

cy 

same  series  y  =  P  (x).  In  this  case,  considered  as  an  integral  func 
tion  in  y  and  with  arbitrary  x,  the  function  G  (x,  y)  has  a  repeated 
factor,  say  Q(xt  y),  which  vanishes  for  y  =  P(x).  Next  suppose 
that  G(xty)  is  decomposed  into  its  irreducible  factors  H^x,  y), 
HI(X>  y}>  '  '  •>  and  give  to  x  such  a  value  xl  that  each  of  these 
functions  is  also  irreducible  when  considered  as  a  function  of  y. 
Furthermore,  since  by  hypothesis  G  (xv  y)  =  0  contains  a  repeated 
root  y=P(xl)t  it  is  seen  that  two  of  the  functions  ffi(xv  y), 
H^(xv  y),  •  -  -,  say  ^  and  H^,  vanish  for  y=P(x^.  And  since 
by  hypothesis  these  functions  are  both  irreducible  with  regard 
to  y,  they  are  identical  except  as  to  a  multiplicative  factor  which 
is  independent  of  y.  But  as  ffl  (x,  y)  and  ff2  (x,  y)  are  identical 
in  y  for  an  indefinitely  large  number  of  values  such  as  x  =  xv  it 
follows  that  the  coefficients  of  like  powers  of  y  in  these  two 


THE  SCHEEFFER  THEORY  59 

functions  are  identical,  so  that  G(x,  y)  is  divisible  at  least  by  the 
square  of  an  integral  function  H(x,  y). 

If  at  least  one  of  the  four  functions,  say  G(x,  ^^(x))t  vanishes 
for  values  of  x  other  than  x  =  0  within  the  fixed  intervals,  while 
for  all  other  values  this  function  retains  the  same  sign,  and  if  the 
other  three  functions  are  invariably  of  this  same  sign,  then  G  (0,  0) 
is  an  improper  extreme  of  G(x,y).  It  follows  that  as  a  necessary 
condition  for  G  (x,  y)  to  have  an  improper  extreme  on  the  position 
x  =  0,  y  =  0,  G  (x,  y)  must  contain  as  factor  the  even  power  of 
an  integral  function  H(x,  y)  which  not  only  vanishes  for  x  =  0, 
y  =  0  but  also  for  values  x,  y  whose  absolute  values  are  arbitrarily 
small.  For  if,  in  accordance  with  the  above  remarks,  G  =  HkG, 
where  G  (x,  y)  contains  no  root  y  =  P  (x)  which  is  also  contained 
in  H(x9  y),  and  if  k  is  odd,  then  as  y  passes  through  the  value 
y  =  P  (x)  the  function  Hk  changes  sign  and  therefore  has  values 
with  opposite  sign. 

Example  1.  Let  f(x,  y}  =  ay*  +  2  bxzy  +  ex*  +  R5  (x,  y},  where  «  >  0  and 
R5  (x,  y}  denotes  any  series  beginning  with  terms  of  the  fifth  order  in  x 
and  y. 

Writing  G  (.r,  y)  =  ay"  +  2  bx*y  +  ex4,  it  is  seen  that  for  x  constant  and 

2\/~*  7i 

j  y  |  =  |  x |,  — —  =  2  (a?/  +  bx2)  is  zero  only  for  y  = x2.    We  thus  have 

„/          b    A      ac-b2    . 
G  [x, x2 }  = F 

\          a     I  a 

and  G  (x,  ±  x)  =  ax'2  ±  2  bx*  +  cx\ 

The    first    expression   offers   the    lower   limit,  while   either  G  (x,  +  x)  or 
G  (x,  —  x)  offers  the  upper  limit. 

We  have  three  cases  to  consider : 

(a)  ac  —  b2  <  0.  Then  of  the  two  limits  one  is  positive  and  the  other 
negative.  It  follows  that  G  (0,  0)  is  not  an  extreme  of  G  (x,  a;),  and  as  both 
limits  begin  with  powers  of  x  not  exceeding  the  fourth,  the  Scheeffer 
theorem  is  applicable,  which  shows  that/(0,  0)  is  not  an  extreme  of  f(x,  y). 

(/?)  ac  —  b2  >  0.  It  follows  since  a  >  0  that  c  must  also  be  positive.  The 
two  limits  just  derived  are  both  positive.  Continuing  we  must  next  deter 
mine  the  other  two  limits.  When  y  is  constant  and  |  x  \  =  |  y  \ ,  we  have  by 
solving  the  equation 

0  =  —  =  4  x  (by  +  ex2) 

CX  : 

the  two  values  x  =  0     and     x  =  ± \! y-. 

\      c 


60  THEOKY  OF  MAXIMA  AND  MINIMA 

If  b  ^  0  the  latter  value  may  be  neglected  (§  38),  since  the  exponent  of  y 
lies  between  0  and  1.    If  &  =70  this  value  coincides  with  the  first. 
We  observe  that  each  of  the  functions 

G  (0,  y)  =  ay*     and     G  (±  y,  y}  =  af  +  2  %3  +  cy* 

is  positive.  It  follows  from  Stolz's  theorem  that  G  (0,  0)  is  a  proper  mini 
mum  of  G  (x,  y}  ;  and  since  the  power  of  x  or  y  on  the  right-hand  side  of 
any  of  the  four  limits  is  not  greater  than  4,  the  Scheeffer  theorem  shows 
that/(0,  0)  is  a  proper  minimum  of  f(x,  y). 

(y)  ac  —  b2  =  0.     From    above  G  (x,  -    —  \  =  0,   while   the  other  three 

limits  are  all  positive.  In  this  case  G  (0,  0)  is  an  improper  minimum  and 
the  Scheeffer  theorem  is  not  applicable,  so  long  as  we  regard  R^  (x,  y~)  as  an 
arbitrary  power  series  with  initial  term  of  the  fifth  or  higher  dimension. 
(Stolz,  p.  235.) 

Example  2.  f(x,  y)  =  y2  +  (ax2  +  2  bxy  +  cy2)  y  +  R±  (x,  y},  (a  *  0). 
We  have  here  G  (x,  y)  =  y2  +  (ax2  +  2  bxy  +  cy2)  y. 

Taking  x  constant  and  \y\  =  \x\,  we  find  as  a  solution  of 

?>.S~*1 

—  =  0  =  2  y  +  ax2  +  2  bxy  +  cy2  +  2  y  (bx  +  cy) 


Forming  the  functions 

G(x,  $(x))=-^-x*  +  ••-     and     G(x,  ±  x)  =  x2  +  [2  b  ±  (a  +  c)];r3 

it  is  seen  that  the  first  furnishes  the  lower  limit,  while  one  of  the  last 
functions  offers  the  upper  limit.  It  is  evident  that  with  x  taken  sufficientlv 
small  these  two  limits  have  contrary  signs,  so  that  G(0,  0)  is  not  an  extreme 
of  G  (x,  y).  Furthermore,  since  the  lower  limit  begins  with  a  power  of  x 
greater  than  3,  the  added  theorem  of  §  31  is  not  applicable. 

Proceeding  further  and  taking  y  constant  and  |a;|5fjyj,  we  have  as  a 
solution  of 

2>/~~i 

—  =  0  =  2  y  (ax  +  by}  (since  y  is  taken  constant) 
x  =  --  ?/,  which  cannot  be  considered  (§  38) 


unless  1  6  1  <  |  a  .    Forming  the  functions 


it  is  seen  that  both  the  upper  and  lower  limits  are  positive.  It  follows  that 
the  added  theorem  is  not  applicable.  We  cannot,  therefore,  make  a  negative 
assertion  regarding  the  extremes  of  f(x,  y}-  (Stolz.) 


THE  SCHEEFFER  THEORY  61 

Example  3.  /(*,  y)  =  f  +  x*y  +  a;4  +  R5(x,  y}. 
In  this  example  we  have    G  (x,  y)  =  y2  +  x2y  +  x4. 
With  .r  constant  and  |  y  \  s  |  #  ,  we  have  as  the  solution  of 


We  thus  have  the  functions 


With  y  constant  and    x  =\y\>  we  have  from 
?£  =  2xy  +  *3*  =  0,     ^ 
It  follows  at  once  that 

==fjfl  and 

The  value  G*(0,  0)  is  consequently  a  proper  minimum  of  G(x,  y),  and  as 
none  of  the  above  series  has  an  initial  term  with  exponent  greater  than  4,  it 
follows  from  Scheeffer's  theorem  thaty(0,  0)  is  a  proper  minimum  of  f(x,  y). 
Although  there  is  a  proper  minimum  for  f(x,  y}  =  y2  +  x^y  +  a;4,  it  may  be 
shown  that  G(x,  y}  =  y2  +  x2y  has  neither  a  maximum  nor  a  minimum. 
(Scheeffer,  loc.  cit.,  p.  573.) 

Example  4.    Peano's  classic  example  : 

/fcy)«0(*j)  +  JW*y)i 

where  G  =  y2  —  (  p2  +  ^2)  a:2?/  +  p2q2x4. 

With  a;  constant  and  |y|=i  ar|,  we  have 


so  that 

Forming  the  functions 


it  is  seen  that  the  upper  limit  is  positive,  while  the  lower  limit  is  negative. 
It  follows  that  (7(0,  0)  is  not  an  extreme  of  G(x,  y}  ;  and  as  the  initial  terms 
on  the  right  have  exponents  that  are  not  greater  than  4,  it  follows  from 
the  Scheeffer  theorem  that/(0,  0)  is  not  an  extreme  of  f(x,  y). 


62  THEOKY  OF  MAXIMA  AND  MINIMA 

Example  5.  /(*,  y)  =  G(x,  y}  +  ^lg(ar,  y), 

where       G(x,  y)  =  *22/4  -  3  a;4/  +  (a?V2  -  3  xy7  +  y8)  -  10  x10/y  +  5  x12. 
With  x  constant  and  |y|  =  |#|,  we  have  from 

fV* 

—  =  4  x*y*  -  9  xY  +  (2  x*y  -  21  xy*  +  8  y7  -  10  a:10)  -  0, 

as  a  solution  (see  §  145), 

y  =  2x2  +  f  x4  +  •  •  •  =  <£(*),  say. 
Forming  the  functions 

G(x,  <£(*))  =  -  4  x10  +  •  •  •     and      G(x,  ±  x)  =  xf<  ±  •  •  ., 

which  (see  again  §  145)  offer  the  upper  and  lower  limits  of  G(x,  y},  it  fol 
lows  from  Stolz's  theorem  and  the  Scheeffer  theorem  that  neither  G(x,  y} 
nor  /(a:,  y)  has  an  extreme  on  the  position  x  =  0,  y  =  0.  (Scheeffer,  loc.  cit. 
p.  575.) 

PROBLEMS 

1.  Show  that/(0,  0)  is  a  minimum  of 

/(*»  30  =  /  +  *6  -  108  x^y  -x*  +  R9  (x,  y}.    (Stolz.) 

2.  Writing  £(*,  y)  =  f  -  2  x*y  +  a:4  +  ?/4, 


show  that  G'(0,  0)  is  a  minimum  for  the  first  function  but  that  /(O,  0) 
is  not  a  minimum  for  the  second  function.  Write  in  the  latter  expression 
y  =  x2.  (Scheeffer.) 

IV.    THE  METHOD  OF  VICTOR  VON  DANTSCHER 

42.  Instead  of  considering  the  extremes  upon  the  straight  lines 
through  the  point  P(xQ9  y0)  we  may  derive  the  criteria  for  maxima 
and  minima  in  the  neighborhood  of  the  points  on  these  lines  on  both 
sides  of  the  point  (#0,  y0)  in  the  ^y-plane.  With  Von  Dantscher* 
let  the  straight  lines  through  (#0,  y0)  be  denoted  by 

(1) 
so  that 


or       -=      x- 


where  \  and  /*  are  real  variables  such  that  X2+  i^=  1  and  where  /o 
is  a  real  variable  which  may  have  both  positive  and  negative  values. 

*See  Math.  Ann.,  Vol.  XLII,  p.  89. 


THE  SCHEEFFER   THEORY  63 

For  extremes  of  f(x,  y)  at  the  point  P§  we  must  have 

/in  case  of  a  maximum\ 


f(xQ  +  \p, 

\proper     or     improper/ 

,.  /  in  case  of  a  minimum\ 

/(«<>»  ^0)  -  ° 

\proper  or    improper/ 

for  all  values  of  p  of  a  certain  interval 

-p<p<q, 

while  for  values  p  =  p  or  p  =  q  the  above  difference  not  only  vanishes 
but  changes  sign. 

The  thesis  of  Yon  Dantscher  may  be  stated  as  follows  :  "  If  the 
lower  limit,  r  say,  of  p  in  the  region  X2+  /n2=l  is  different  from 
zero,  then  f(xQ,  y0)  is  a  maximum  or  minimum  for  the  surface- 
neighborhood  of  the  point  (x0,  y0);  but  if  the  lower  limit  of  p  is 
zero,  then  on  the  position  XQ,  yQ  there  is  neither  a  maximum  nor 
a  minimum  of  the  function  f(xy  y)" 

The  decision  as  to  whether  a  maximum  or  minimum  exists  for 
a  given  function  f(x,  y)  on  a  point  #0,  yQ  in  whose  neighborhood 
f  (x,  y)  can  be  developed  in  integral  positive  powers  of  x  —  #0  =  h  , 
y  —  yQ  =  A-,  and  on  which  point  the  first  partial  derivatives  with 
respect  to  x  and  y  both  vanish,  is  consequently  reduced  to  the 
investigation  as  to  whether  the  quantity  p  is  different  from  zero 
or  not. 

If  in  the  supposed  development  the  ?ith  dimension  is  the  first 
whose  terms  do  not  all  vanish,  we  write 

(2)  f(zQ+Ji,yQ+*)-f(xQ,y0) 

=  g(h,  Jc)  =  (h,  k)n  +  (h,  *)n+l+(h,  &),I  +  2+  •  •  •,    (n  ^  2) 

where  (h,  k)n  denotes  the  sum  of  the  terms  of  the  ?ith  dimension 
in  h  and  &,  etc. 

If  we  write  in  this  expression 

(3)  h  =  \p,     k  =  np,     \*+fjfi=l, 
we  have 

(4)  ^,*)=^(X,M)wH-p(X,/*)w+1-h---]  =  ^(p;  X,  /x). 


64  THEORY  OF  MAXIMA  AND  MINIMA 

The  factor  pn  may  be  omitted,  since  to  the  value  p  =  0  there 
corresponds  the  position  h  =  0,  k  =  0  itself.  The  quantity  r  is 
accordingly  nothing  other  than  the  lower  limit  of  the  absolute 
values  of  the  real  roots  of  the  equation 

(5)  </>(/>;  X,  /*)  =  (\,  /*)W  +  (X,  /*)w  +  1/>+...=  0. 

From  this  the  following  is  at  once  evident  : 

CASE  I.  If  (h,  Jc)n  is  a  definite  form  (§13),  that  is,  one  which 
takes  the  value  zero  for  the  one  and  only  pair  of  values  h  =  0, 
lc  =  0,  which  case  can  only  enter  when  n  is  even,  then  (X,  p)n 
is  different  from  zero  for  all  values  X,  /*  which  are  different 
from  zero,  and  consequently  |(X,  fi)n\  has  a  lower  limit  I 
which  is  different  from  zero.  We  may,  consequently,  for  the 
region  X2  -f  y?  =  1,  determine  a  positive  quantity  r  such  that 
for  |  p  |  <  T  we  have 

(X,  p)nsi>\(\,  /*)„+!/>  +  (x,  ?)»+»?+  •••• 

The  equation  (5)  has  therefore  no  root  p  whose  absolute  value  is 
not  greater  than  r  ;  the  quantity  r  is  therefore  different  from  zero, 
and  there  is  consequently  a  maximum  or  minimum  according  as 
(h,  k)n  is  a  negative  or  positive  form. 

CASE  II.  If  (h,  k)n  is  an  indefinite  form  (§  13),  that  is,  one 
which  for  real  pairs  of  values  (h,  k)  takes  both  positive  and  nega 
tive  values,  then  also  (X,  ji)n  is  such  a  form.  It  is  then  easy  to 
show  that  in  this  case  the  equation 


in  any  interval  as  small  as  we  please  —  e  <  p  <  e,  has  roots  that 
are  different  from  zero,  and  consequently  r  =  0  and  also  f(xQ,  y0) 
is  neither  a  maximum  nor  a  minimum. 

43.  CASE  III.  We  come  finally  to  the  semi-definite  case 
(§  13)  ;  that  is,  one  where  (h,  Jc)n  vanishes  for  pairs  of  values 
h,  k  which  are  different  from  zero,  but  does  not  change  sign. 
It  contains  necessarily  real  linear  factors,  and,  in  fact,  each 
one  to  an  even  power.  The  number  n  is  consequently  even, 


THE  SCHEEFFER  THEORY  65 

and  it  follows   tha,t   (X,  /z)?l  is   necessarily  also  a  semi-definite 
form,  whose  factors  are,  say, 

(6)  kfi  -  hj,  kji  -  hzk,  ...,kmh-  hmk, 

so  that  (h,  k)n  is  of  the  form 
(h,  k}n  =  (kji  -  ^ 


where  llf  /2,  •  •  .  lm  are  positive  integers  and  (h,  *)»-2(«1+  !,+  • 
is  a  definite  form  or  a  constant. 

To  each  such  linear  factor  kji  —  hak(<r  =  1,  2,  -  •  •  ,  m)  of  (h,  k)n 
there  corresponds  a  linear  factor  fjL0\  —  \ap  of  (X,  /*)„,  where 


with  arbitrary  sign  of  V  ^  4-  ^>  si^ce  this  constant  enters  only 
to  squared  terms  in  (A,  k)n.  If  X,  /*  approach  a  pair  of  values 
Xff,  /JL0.  for  which  (X,  /j,)n  vanishes,  then  of  the  roots  of  the  equation 

<£(/>;  \  /*)  =  (>.,  /*)»  +  (^  A*)M  +i/»  H  ----  =  o, 

one  or  several  become  indefinitely  small. 

Of  course  we  may  exclude  the  case  where  all  the  quantities 
(X,  fi)n  +  v(v^1)  simultaneously  vanish;  for  then  <£(/>;  X,  /i)  =  0 
for  every  arbitrary  small  value  of  p,  and  consequently  /(#0,  y0) 
is  neither  a  maximum  nor  a  minimum. 

We  have  next  to  see  whether  among  the  roots  of  cf>(p  •  X,  p)  —  0, 
which  become  indefinitely  small  when  (X,  p}n  becomes  indefinitely 
small,  there  are  real  roots  or  not.  If  no  real  roots  appear,  then 
r  >  0  and  f(%Q,  y§)  is  a  maximum  if  the  semi-definite  form  (h,  k)nf 
when  it  does  not  vanish,  is  negative,  while  it  is  a  minimum  if  (h,  k)n 
is  positive. 

When  there  appear  real  roots  the  investigation  may  be  carried 
out  as  follows  :  In  order  to  consider  the  function  <£>  (p  ;  X,  /*)  in  the 
neighborhood  of  the  point  \ff,  fjLff,  we  write  ' 

(7)  X  =  X<7+i/,      fjL  =  nff+v, 

where  u  and  v  are  variable  quantities. 


66  THEORY  OF  MAXIMA  AND  MINIMA 

Since  X2  +  //<2=  1  and  \*  +  p%  =  1,  we  must  have 

^2+^2+  2X^  +  2/4^  =  0, 

where  it  is  certain  that  one  of  the  quantities  \  or  pv  is  different 
from  zero. 

If  /^^  0  we  have  at  once  from  the  equation  just  written 


*  -  (2  Xaw  +  w2), 

where  the  positive  sign  is  taken  with  the  root,  since  from  (i)  u  and 
v  vanish  simultaneously.    Further,  noting  the  development 


it  is  seen  that 

(8)  „  =  _*-„_     l^i-A^s  ----  ; 

Pa  2  /*  8  2  /1  6 

and  if  X,  =£  0, 

(9)  «  =  -?»-  ^i*"-"" 

'V  2  Ag. 

Writing  these  values  in  </>  (p  ;  X,  /x),  we  have 

—  u  ----  ^ 

p+ 


«+! 

and  v 


n  +  1 

which  for  sufficiently  small  values  of  \u\  and  p  or  of  v|  and  |/o| 
are  certainly  convergent  and  may  be  arranged  in  powers  of  u 
and  p  or  of  v  and  /a. 

Since  (Xa,  /*<r)w=  0,  it  is  seen  that  <k(0,  0)=  0;  the  case  that 
^(0,  p)  vanishes  identically  may  be  excluded,  as  has  already 
been  remarked. 

If  pp  is  the  lowest  power  of  p  in  </>ff(0,  p),  the  equation  ^(O,  p)  =  0 
has  exactly  p  roots  p,  which  become  indefinitely  small  with  u  or  v. 
We  must  next  see  whether  there  are  real  roots  among  these  p  roots. 


THE  SCHEEFFER  THEORY  67 

If  the  equation  <f>ff(u,  p)=  0  has  no  real  root  p  which  becomes 
indefinitely  small  with  u  or  v,  then  for  any  arbitrarily  small  posi 
tive  quantity  e  a  positive  quantity  8  cannot  be  found  so  small 
that  in  the  interval  —  e  <  p  <  e  there  is  situated  a  root  p  of 
<j>0(ut  p}—  0  or  of  (f><r(v>  P)  which  is  different  from  zero  and  which 
belongs  to  a  value  u  or  v  in  the  interval  —  8 <  %  <  8  or  —  &<v  <8. 
Hence  there  exist  positive  quantities  8  and  e  so  small  that  the 
function  fa  which  vanishes  simultaneously  with  u  and  p  or  with 
v  and  p  in  the  region 

—  8  <  %  <  8     or     —  8  <  v  <  8,     —  e  <  p  <  e 

takes  values  that  are  different  from  zero  on  every  position  u,  p 
or  v,  p  which  is  different  from  0,  0,  and  these  values  have  neces 
sarily  the  same  sign.  For  if  4>a(u',  p')>Q  and  <t>ff(u",  p")  <  0, 
then  with  a  continuous  passage  from  the  position  ur,  p'  to 
the  position  u",  p" ,  which  both  lie  within  the  interior  of  the 
realm  in  question  and  which  passage  does  not  pass  through  the 
position  0,  0,  there  must  be  a  position  u0,  p0  at  which  fa(u,  p) 
vanishes ;  but  there  are  no  such  positions.  It  follows  that 
<£0.(0,  0)  is  itself  a  maximum  or  minimum  provided  the  equa 
tion  $v(u,  p)=  0  has  no  real  root  which  becomes  simultaneously 
indefinitely  small  with  u  or  v.  Inversely,  it  is  also  true  that  if 
(^(0,  0)  is  a  maximum  or  minimum  of  <j>v(ut  p),  the  equation 
4>ff(u9  p)=0  has  no  real  root  which  becomes  indefinitely  small 
with  u  or  v. 

If,  on  the  other  hand,  the  equation  <f>ff(uf  p)=  0  has  real  roots 
which  become  indefinitely  small  with  u  or  v,  then  $ff(0,  0)  is 
neither  a  maximum  nor  a  minimum ;  and  vice  versa,  if  $ff(0,  0) 
is  not  a  maximum  or  minimum,  then  in  every  region  as  small 
as  we  wish  —8<u<8  or  —8<v< 8,  —  e<p<e  there  are  posi 
tions  u,  p  or  v,  p  which  are  different  from  zero  and  for  which 
<f>0(ut  p)  or  fa(v,  p)  are  zero. 

Through  the  above  consideration  the  criterion  whether  the 
equation  <f>(T=  0  has  or  has  not  real  roots  which  become  indefi 
nitely  small  with  u  or  v  is  reduced  to  the  investigation  whether 
^(0,  0)  is  a  maximum  or  minimum  of  $0(ut  p)  or  <f>ff(v,  p)  or  not. 


68  THEORY  OF  MAXIMA  AND  MINIMA 

We  have,  therefore,  to  apply  the  criteria  of  Cases  I  and  II  of 
§  42  ;  that  is,  to  arrange  <£a  in  dimensions  of  u  and  p  or  of  v  and 
p  and  to  see  whether  the  terms  of  lowest  dimension  form  a  definite 
or  indefinite  form. 

This  same  process  must  be  applied  to  each  of  the  m  real  linear 
factors  ^ff\  —  \IA  (a-  =  1,  2,.-.,  ra)  that  are  different  from  one 
another  (p.  65),  it  being  evidently  sufficient,  since  u  and  v  become 
simultaneously  indefinitely  small,  for  those  linear  factors  in  which 
A^  and  /^  are  both  different  from  zero  to  consider  only  one  of 
the  functions  $„(&,  p)  or  <f)ff(v,  p). 

44.  We  have,  then,  the  following  rule  for  Case  III  : 

If  the  developments  of  the  functions  ^(w,  p),  </>2(w,  p),  •  •  •, 
<t>m(u,  p)  all  begin  with  definite  forms,  then  f(xQ,  y0)  is  a  max 
imum,  when  the  semi-definite  form  (h,  k)n  ^  0,  while  it  is  a  min 
imum  if  (h,  k)n  =  0. 

If  only  one  of  the  functions  $a  (u,  p)  begins  with  an  indefinite 
form,  then  f(xQ)  yQ)  is  neither  a  maximum  nor  a  minimum. 

The  case  remains  undetermined  if  among  all  the  functions 
(j)^  (u,  p)  none  of  them  begins  with  an  indefinite  form,  while  one  or 
several  of  them  begin  with  a  semi-definite  form. 

In  this  case,  for  every  such  function  the  above  process  must 
be  again  applied.  We  do  not  affirm  that  by  using  this  method 
a  determination  may  among  all  conditions  be  made;  but  Von 
Dantscher  says  "  if  the  method,  which  has  been  developed  to  see 
whether  a  series  g(h,  k)  which  begins  with  a  semi-definite  form 
has  or  has  not  on  the  position  h  =  0,  k  =  0  a  maximum  or  mini 
mum,  fails,  the  function  g(h,  k)  contains  an  even  power  of  a 
series  P  (h,  k)  which  vanishes  for  real  pairs  of  values  h,  k  in  every 
region  arbitrarily  small  0<  h  <  8,  0<\k\<8"  (see  §  41), 

Example  1.    Peano's  classic  example  : 

g  (h,  k)  =  k*-  (p2  + 
We  have  here 


The  semi-definite  form  p?  has  the  linear  factor  /x  so  that  either 
Xx  =  1,     fjil  =  0,     or     Xj  =  -  1,     /*x  =  0. 


THE  SCHEEFFER  THEORY  69 

The  corresponding  values  of  X  and  /*  are  (see  [7]  and  [9]) 


so  that  <k  (r,  p)  -  r*  -  Q?2  +  ^)  ly> 

The  terms  of  the  second  dimension  in  v  and  p  form  an  indefinite  quad 
ratic  form,  so  that  #(0,  0)  is  neither  a  maximum  nor  a  minimum. 

Example  2.    Let     g  (h,  Jc}  =  -  W  (h  -  kf  +  2  lik*  -  5  WL*  + 

+  h*L*  -  7  W  +  6  h*k  -  10  hs  + 

+  3  7^-4  £8+  .... 
\\  e  then  have 

<£  (p;  X,  /A)  =  -  XV(X-/*)2  +  (2  V6-  5  A2/*6+  3  *V4  +  A.V-7XV+  6  X»p 
+  (-  10  A8  +  XV  +  3  XV4  -  4  /x8)^2  +  .  ... 

We  have  here  to  consider  the  three  linear  factors 

/xxX  —  Xj/x  =  X,     fj^X  —  Xo/x  =  fi,     /x3X  —  X3/A  =  X  —  //.. 
It  follows  that 

X^O,     ^=1;        X.^-1,     ^=0;     X3  =  4='     ^  =  ^=' 

V2  V2 

To  these  values  correspond  the  expressions  : 

X=«,        fJL  =  l-±U2  ----  , 

fji  =  v,      X  =  -  1  +  £  r2  +  •  •  ., 

1  1 

X  =  —  +  w,     p.  =  —  -u  ----  . 

We  thus  have 


^3(u'  p)  =  -  w2+  IWP-  ip2+  •••• 

It  is  seen  that  all  three  of  the  functions  <£  begin  with  definite  quad 
ratic  forms.  The  semi-definite  initial  form  is  negative  when  it  is  not 
zero;  and  accordingly  g  (0,  0)  is  a  maximum.  (Von  Dantscher,  Math.  Ann., 
Vol.  XLII,  p.  100.) 

PROBLEMS 

1  .  Show  that  g  (0,  0)  is  neither  a  maximum  nor  a  minimum  of  the  function 

g  (h,  k)  =  h*P  -  3  A**8  +  h*k*  -  3  hk7  +  k*  -  10  hlok  +  5  Au. 
(See  Ex.  5,  p.  62.) 

2.  Apply  this  method  to  Ex.  3,  p.  61. 

3.  If   22  =  a2  —  x-  —  yz  +  (x  cos  a  +  y  sin  a)2,   find  maximum  and  mini 
mum  values  of  z  and  give  the  geometric  interpretation. 

4.  If  z2  =  2  a  Va:2  +  y2—  x2  +  y2,  find  maximum  and  minimum  values  of  z  ; 
show  that  there  are  improper  extremes  and  give  geometric  signification. 

5.  Find  minimum  value  of  u,  where  u  =  (x2  +  y2)^. 


70  THEORY  OF  MAXIMA  AND  MINIMA 

V.    FUNCTIONS  OF  THREE  VARIABLES 

TREATMENT  IN  PARTICULAR  OF  THE  SEMI-DEFINITE  CASE 

45.  The  theorems  and  proofs  given  by  Stolz  and  Scheeffer  for 
functions  of  two  variables  may  be  extended  at  once  to  functions 
of  three  or  more  variables.  For  example,  f(xQt  y0,  ZQ)  is  a  proper 
maximum  of  f(x,  y,  z)  if  a  positive  quantity  £  can  be  so  deter 
mined  that  for  all  systems  of  values  f,  77,  f,  whose  absolute  values 
are  smaller  than  S  (excepting  f  =  0  =  1;  =  f),  we  have 


If  the  partial  derivatives  of  f(x,  y,  z)  have  definite  values  at 
every  position  of  a  fixed  realm  R,  the  coordinates  XQ,  yQ,  ZQ  of 
those  positions  (if  any)  in  R  which  offer  extremes  of  the  function 
f(x,  y,  z)  must  satisfy  the  equations 


To  apply  the  Stolzian  theorem  we  observe,  if  we  limit  ourselves 
to  a  position  XQ  =  0  =  yQ  =  ZQ,  that  the  collectivity  of  positions 
x,  y,  z  for  which  x\,  \y\,  \z  are  less  than  8  are  distributed  into 
three  kinds  of  realms  :  * 

(1)  always  with  |a?|<8,  x  constant,  and 

\y\S\x\,      \z\S\x\;  "•         - 

(2)  with  y  constant  and  |y|<8,  where  also 

\x\S\y\,     \zs\y\; 

(3)  with  z  constant  and  |z|<8  and 

|*|  S  |*|,      \y\S\,\. 

To  apply  the  Scheeffer  theorem  we  must  consider  the  difference 
f(x,  y,  z)-/(0,  0,  0)=  Gn(x,  y,  z)+Rn+l(x,  y,  z), 

as  in  §  30. 

The  case  where  Gn(x,  y,  z)  is  a  definite  or  indefinite  form  is 
treated  fully  in  Chapter  V. 

*  Stolz,  loc.cit.,  p.  237. 


THE  SCHEEFFER  THEORY  71 

46.  The  case  where  Gn(xy  y,  z)  is  a  nonhomogeneous  fuuctiou 
in  which  the  terms  of  the  lowest  dimension  constitute  a  semi- 
dennite  form  may  be  treated  in  a  manner  analogous  to  that 
given  in  §§  37~41,  as  follows: 

"We  first  determine  the  upper  and  lower  limits  of  G(x,  y,  z) 
with  constant  x  and  |y|  =  4  |«|  =  |4  Geometrically  interpreted, 
this  realm  constitutes  a  square  whose  center  is  the  origin  and 
whose  sides  are  parallel  to  the  y-axis  and  the  z-axis,  the  length 
being  2  14 

The  positions  at  which  G(x,  y,  z)  reaches  one  of  its  limits  may 
lie  (1)  on  the  vertices,  or  (2)  on  the  sides,  or  (3)  within  the 
interior  of  the  square. 

We  have,  consequently,  to  form  the  expressions  corresponding 
to  the  four  vertices 

G(x,  x,  x),  G(xt  x,  —  x),  G(x9  —  x,  x),  G(x,  -  x,  -  x}.  (i) 

For  points  on  the  sides  we  have  to  solve  for  y  the  equations 


cy  dy 

^  for  z  the  equations 


•dz  cz 

Let  the  solutions  of  the  equations  (a}  be 

y  =  3(*)>    y  =  %(*)>   ••• 

and  let  the  solutions  of  (/8)  be 

z=Ql(x],     z  =  Q2(x),     .... 

Those  functions  P  (x)  and  Q  (x)  which  cause  y  and  z  to  fall  without 
the  given  square  are  to  be  neglected  (cf.  §  38). 

With  the  remaining  functions  we  form  the  expressions 

G  (x,  Pl  (x),  x),  .  .  .  ;      G  (x,  x,  Ql  (x)).  (ii) 

For  the  points  within  the  square  we  have  to  determine  y  and  z 
in  terms  of  x  from  the  equations 


=  0 
oy  cz 


72  THEORY  OF  MAXIMA  AND  MINIMA 

If  we  eliminate  z  from  these  two  equations,  we  may  express  y  as 
power  series  in  x  without  constant  term,  say  y  =  fa  (x)y  y  =  </>2'(#), 
•  •  •  (§  29).  To  each  such  power  series  for  y,  say  y  =  <£(#),  there 
corresponds  one  for  z  in  terms  of  xt  say  z  =  \(x),  which  two  series 
written  in  the  two  equations  (7)  cause  them  to  vanish  identically. 
With  these  values  of  y  and  z  we  form  the  expressions 

G(x,  ^(x),  \(x)),     G(xt  02(ar),  X2(^)).  (iii) 

Among  all  the  functions  that  are  found  in  (i),  (ii),  and  (iii)  we 
are  now  able  to  determine  those  two  which  offer  the  upper  and 
lower  limits  of  the  function  G(xt  y,  z)  within  the  interval  in 
question.  These  limits  may  be  denoted  by  G(x,  <&%(x),  A2(#)) 
and  G(x,  4>1(^1),  A^)). 

If,  next,  we  take  y  constant  and  |$|  s&jy  |,  z  |  =  |  ?/  1,  we  may  derive 
in  a  similar  manner  the  upper  and  lower  limits  G(W2(y),  y,  M2(y)) 
and  G^V^y),  y,  M^y)).  Finally,  with  z  constant  and  |#|^|«|, 
|  y  |  ^  |  z  ,  we  derive  the  upper  and  lower  limits 

,  fia(2),  z)     and     ^(N^),  Q^z),  z). 


The  Stolzian  and  Scheefferian  theorems  are  at  once  applicable 
to  these  six  functions  in  three  variables,  the  method  of  procedure 
being  an  evident  generalization  of  these  theorems  for  the  functions 
in  two  variables. 

PROBLEMS 

1.  Make  the  extension  and  generalization  of  Von  Dantscher's  method 
to  the  treatment  of  functions  in  three  variables. 

2.  In  the  line  of  intersection  of  two  given  planes  find  the  nearest  point 
to  the  origin  of  coordinates. 


CHAPTER  V 

MAXIMA  AND  MINIMA  OF  FUNCTIONS  OF  SEVERAL  VARIABLES 
THAT  ARE  SUBJECTED  TO  NO  SUBSIDIARY  CONDITIONS 

I.   ORDINARY  EXTREMES 

47.  It  will  be  presupposed  in  the  following  discussion,  unless 
it  is  expressly  stated  to  the  contrary,  that  not  only  the  quantities 
that  appear  as  arguments  of  the  functions  but  also  the  functions 
themselves  are  real,  and  that  the  functions,  as  soon  as  the  vari 
ables  are  limited  to  a  definite  continuous  region,  have  within  this 
region  everywhere  the  character  of  one-valued  regular  functions. 
Regular  functions  are  defined  in  the  following  manner :  A  func 
tion  f(x)  is  regular  within  certain  fixed  limits  of  x  if  the  func 
tion  is  defined  for  all  values  of  x  within  these  limits  and  if  for 
every  value  a  of  x  within  these  limits  the  development 

f(a  +  h)  =/(«)  +  £/'<«)  +  ^/"  («)+... 

is  possible  ;  the  series  must  be  convergent  and  must  in  reality  (see 
§  136),  represent  the  values  of  the  function  within  this  neighborhood. 

In  other  words :  A  function  f(x)  is  regular  in  the  neighbor 
hood  of  the  position  x  —  a  if  the  function  in  this  neighborhood 
has  everywhere  a  definite  value  which  changes  in  a  continuous 
manner  with  x.  (Cf.  Weierstrass,  Werke,  Vol.  II,  p.  77.) 

A  one-valued  analytic  function  f(xv  x%,  •  •  •,  xn)  of  several  vari 
ables  behaves  regularly  on  a  definite  position  (x1=  av  x2=  a2,  •  .  >, 
xn=  an)  if  in  the  neighborhood  of  this  position  we  may  express 
the  function  through  a  series  of  the  form 

^A^  »v-  •  •>  *n(xi-  ai)Vl(x2-  azY*  '  '  •  (xn~  ««)""> 
where  vl}  z>2,'«  •  .,  vn  are  positive  integers  or  zero,  and  where  the 
coefficients  Av^  v>,  •  •  .,  Vn  are  quantities  that  are  independent  of  the 
variables.    (Cf.  Weierstrass,  Werke,  Vol.  II,  p.  164.) 

73 


74  THEORY  OF  MAXIMA  AND  MINIMA 

The  discussion  is  thus  limited  to  such  functions  as  are  analytic 
structures  of  the  nature  described  more  in  detail  in  §§  130,  131. 
Only  for  such  functions  can  we  derive  general  theorems,  since 
for  other  functions  even  the  rules  of  the  differential  calculus 
are  not  applicable ;  in  other  words  we  shall  consider  only  the 
ordinary  extremes. 

The  problem  of  finding  those  values  of  the  argument  of  a 
function  f(x)  for  which  the  function  has  a  maximum  or  mini 
mum  value  is  not  susceptible  of  a  general  solution,  for,  besides 
the  cases  of  the  extraordinary  extremes  of  §§  5-7,  there  are  func 
tions  which,  in  spite  of  the  fact  that  they  may  be  defined  through 
a  simple  series  or  through  other  algebraic  expressions  and  which 
vary  in  a  continuous  manner,  have  an  infinite  number  of  maxima 
and  minima  within  an  interval  which  may  be  taken  as  small 
as  we  wish.*  Such  functions  do  not  come  under  the  present 
investigation. 

48.  We  say  (see  §  1)  that  a  function  f(x)  of  one  variable  lias 
a  proper  maximum  or  a  proper  minimum  at  a  definite  position 
x  =  a  if  the  value  of  the  function  for  x  =  a  is  respectively  greater 
or  less  than  it  is  for  all  other  values  of  x  which  are  situated  in 
a  neighborhood  x  —  a  <  S  as  near  as  we  wish  to  a. 

The  analytical  condition  that  f(x)  shall  have  for  the  position 
x  =  a 


a  proper 
a  proper 


maximum,  is  expressed  by  f(x)—f(a)  <  0  \         ,  ., 

minimum,  is  expressed  by  f(x)  —/(a)  >  0  J 


In  the  same  way  we  say  a  function  f(xv  x2,  •  •  •,  xn)  of  n 
variables  has  at  a  definite  position  xl  =  av  x2  =  a2,  -  -  .,  xn=  an 
a  proper  maximum  or  a  proper  minimum  if  the  value  of  the 
function  for  x1  =  av  x2=  «2,  •  •  .,  xn=  an  is  respectively  greater 

*  A  function  may  have  in  an  interval  as  small  as  we  wish 

(1)  an  infinite  number  of  discontinuities, 

(2)  an  infinite  number  of  maxima  and  minima, 

and  still  be  expressed  through  a  Fourier  series. 

See,  for  example,  H.  Hankel,  Ueber  die  unendlich  oft  oscillirenden  und  unstetigen 
Functionen  (Tubingen,  1870);  Lipschitz,  Crelle,  Vol.  LXIII,  p.  296:  P.  du  Bois- 
Reymond,  Abh.  der  Bayer.  Akad.,  Vol.  XII,  p.  8,  and  also  same  volume,  Part  II, 
Math.-Phys.  Classe  (1876). 


NO  SUBSIDIARY  CONDITIONS  75 

or  less  than  it  is  for  all  other  systems  of  values  situated  in  a 
neighborhood  ^..^  (X==  lf  2,  ...,») 

as  ?iear  as  we  i0is&  to  the  first  position]  and  the  analytical 
condition  that  the  function  f(xv  x2,  •  •  •,  xn)  shall  have  at  the 

position  #!  =  av  x2  =  a2,  •  •  •,  xn=  an 

a  proper  maximum,  isf(xv  x%,  •  •  •,  <£n)—f(ai>  ai>  '  '  •>  fln)  <  0, 
a  proper  minimum,  isf(xv  x2,  •  •  •,  xn)  —  f(av  a2,  •  •  •,  an)  >  0, 

for  |a;A—  aA|<8x  (X  =  1,  2,..-,  ?i),  where  the  quantities  SA  are 
arbitrarily  small.  Improper  extremes  take  the  place  of  the 
proper  extremes  above  when  we  allow  the  equality  sign  to 
appear  with  the  inequality  sign,  as  in  §  1. 

49.  The  problem  which  we  have  to  consider  in  the  theory 
of  maxima  and  minima  is,  then,  to  find  those  positions  at 
which  a  maximum  or  minimum  really  enters. 

We  shall  give  a  brief  resume  of  this  problem  for  functions 
of  one  variable  and  then  make  its  generalization  for  functions 
of  several  variables. 

If  xv  x2  are  two  values  of  x  situated  sufficiently  near  each 
other  within  a  given  region,  then  the  difference  of  the  corre 
sponding  values  of  the  function  is  expressible  in  the  form  : 


where  6  denotes  a  quantity  situated   between   0   and    1  ;    or,  if 
x1  is  written  equal  to  x  and  x^x  +  h, 

[1]  f(x  +  h)-f(x)  =  hf  (x  +  0  h). 

From  this  theorem  may  be  derived  Taylor's  theorem  in  the 
form,* 

[2]         f(x  +  h)  -f(x)  =  hf'(x)  +  1  A*/"  (*)+••' 

--«  (x)  +  l  h*fW  (x  -h  0h). 


(71  —  1)1  n. 

*  See  Jordan,  Cours  D'  Analyse,  Vol.  I,  §§  249-250. 


76  THEORY  OP'  MAXIMA  AND  MINIMA 

In   the  two  formulae  last  written,  instead  of  x  +  h  write  x 
and  write  a  in  the  place  of  x\  they  then  become 

[  1°]  /(*)  -  /(«)  =  (*-« 

and 

[2"]      f(x)-/(a) 


Since  f(x)  is  a  regular,  and  consequently  continuous,  function, 
the  same  is  true  of  all  its  derivatives.  If  f'(a)  is  different 
from  zero,  then  with  small  values  of  h  =  x  —  a  the  value  of 
/'  (a  +  Oh)  is  different  from  zero  and  has  the  same  sign  as  /'  (a). 

According  to  the  choice  of  hy  which  is  arbitrary,  the  differ 
ence  /(#)  —  /(»)  can  be  made  to  have  one  sign  or  the  opposite 
sign,  if  /'  (a)  is  either  a  positive  quantity  or  a  negative  quantity. 
Hence  the  function  f(x)  can  have  neither  a  maximum  nor  a  mini 
mum  value  at  the  position  x  =  a  if  f  (a)  ^  0. 

We  therefore  have  the  theorem  :  Extremes  of  the  function 
f(x)  can  only  enter  for  those  values  of  x  for  which  f  (x) 
vanishes  (see  §  2). 

It  may  happen  that  for  the  roots  of  the  equation  /'  (x)  =  0 
some  of  the  following  derivatives  also  vanish.  If  the  nih  deriva 
tive  is  the  first  one  that  does  not  vanish  for  the  root  x=a,  then 
from  equation  [2a]  we  have  the  formula 


f(x)  -/(a)  =0(a.  +  0(a!  -  a)}, 

and  with  small  values  of  h  =  x  —  a,  owing  to  the  continuity  of 
/(")(£),  the  quantity  /<">(«  +0h)  will  likewise  be  different  from 
zero  and  will  have  the  same  sign  as  /(n)  (a).  If,  therefore,  n  is 
an  odd  integer,  we  may  always  bring  it  about,  according  as  h  is 
taken  positive  or  negative,  that  the  difference  f(x)—f(a)  with 
every  value  of  f^  (a)  has  either  one  sign  or  the  opposite  sign  ; 
consequently  the  function  f(x)  will  have  at  the  position  x  =  a 
neither  a  maximum  nor  a  minimum  value. 


NO  SUBSIDIARY  CONDITIONS  77 

If,  however,  n  is  an  even  integer,  then  hn  is  always  positive, 
whatever  the  choice  of  h  may  have  been  ;  consequently  the 
difference  f(x)  —  f(a)  is  positive  or  negative  according  as  /(n)(a) 
is  positive  or  negative. 

In  the  first  case  the  function  f(x)  has  a  minimum  value  at 
the  position  x  =  a  ;  in  the  latter  case,  a  maximum. 

Taking  this  into  consideration  we  have  the  following  theorem 
for  functions  of  one  variable  (§  3)  : 

Extremes  of  the  function  f(x)  can  only  enter  for  the  roots  of 
the  equation  f  (x)  =  0.  If  a  is  a  root  of  this  equation,  then  at  the 
position  x  =  a  there  is  neither  a  maximum  nor  a  minimum  if  the 
first  of  the  derivatives  that  does  not  vanish  for  this  value  is  of 
an  odd  degree  ;  if,  however,  the  degree  is  even,  then  the  function 
has  a  maximum  value  for  the  position  x  =  a  if  the  derivative 
for  x  =  a  is  negative,  a  minimum  if  it  is  positive. 

50.  To  derive  the  analog  for  functions  of  several  variables,  we 
start  again  with  the  Taylor-Lagrange  theorem  *  for  such  functions. 
This  theorem  may  be  derived  by  first  writing  in  f(xlt  x2,  >  •  •,  xn) 

«A  =  «A  4-  u(xx  -  «A),     (X  =  1,  2,  •  .  .,  n), 

where  u  is  a  quantity  that  varies  between  0  and  1  ;  we  then  apply 
to  the  function 

<l>(u)  =  f(a1+u(x1-al),  aa+tA(a^-a2),  •  •  .,  an+u(xn-an)) 
Maclaurin's  theorem  for  functions  of  one  variable,  viz.: 

[3]  f(«) 


and,  finally,  in  this  expression  write  u  =  l,  as  follows  : 

For  brevity  denote  by  fk(xv  x2,  -  •  .,  xn)  the  first  derivative  of 
f(xv  xz,  •  •  -,  xn)  with  respect  to  xk  and  by  fkl,^(xv  x2,  .  .  .,  xn) 
the  derivative  of  f(xl,  #2,  •  •  -,  xn)  with  respect  to  xki  and  xkt, 

thatis'    f     ,,  ay^,^,...,^,.) 

A.nft>4.—  .*)•         gXkgXki 

*See  Lagrange,  The'orie  des  Fonctions,  p.  152. 


78  THEOKY  OF  MAXIMA  AND  MINIMA 

It  follows,  then,  that 


kvk2,^,k>n-i  '  \ 

Hence,  from  [3]  we  have 

(f>(u)-f(av  aa,  .  .  .,  an)=  ^^{ffc(alf  aa>  .  .  .,  an)(xt-at)} 

*•    /• 


«/Wl 

^1^  ^ 


an+  6u(xn-  an}}(xki-  aki)  .  .  .  (xkm-  akm)} 
From  this  it  follows,  if  we  write  u  =  1,  that 
f(xv  a?a,  .  .  .,  a;w)  -/(«!,  «2,  •  •  -,  «„)=]£{/*(«!,  «2,  -  -  .,  aM)(ajfc-afc)} 

/•; 

+  2!  S  ^A,  *i(ai>  V  "  "• 

*  /*p  /."2 
+ 


_ 

/  '  Ajj,  A*2,  •  •  •  ,  km  _  i 

(%«_!-%,„_,) 

2       {A3*2,...,;u(a1  +  <%1-  «!>,  «2  +  l9(^2-  a2),  .  .  ., 
'  *t,*n  -V*. 


NO  SUBSIDIARY  CONDITIONS  79 

51.  We  are  not  accustomed  to  Taylor's  theorem*  in  the  form 
just  given  ;  to  derive  this  theorem  as  it  is  usually  given,  observe 
that  upon  performing  the  indicated  summations  each  of  the  in 
dices  kv  k2,  -  -  .,  independently  the  one  from  the  other,  takes  all 
values  from  1  to  ?i,  so  that  the  Xth  term  in  the  development  is 
a  homogeneous  function  of  the  Xth  degree  in  x1  —  av  x2  —  a2, 
•  •  -,xn  —  an.  The  general  term  of  this  homogeneous  function  may 
be  written  in  the  form 

i  D  -  .V  •  fa  -  a^(x2  -  «2)*.  •  •  •  (xn  -  an)^ 
where  Xj  +  X2  +  •  •  •  +  X,,  =  X, 

D  is  the  definite  differential  quotient 

_  /(A1+A.+  .-.  +  X«)/«       fl  n    \ 

f  (l   2" 


and  N  is  the  number  of  permutations  of  X  elements  of  which  \v 
X2,  •  •  •,  \n  respectively  are  alike  ;  that  is, 

"^l  X2!   ...Xj' 

Furthermore,  writing  xk  —  ak  =  hk,  we  have,  finally, 
[4]     f(xv  x2,  •  .  .,  xn)-f(av  a2,  .  .  .,  an) 


*Stolz  (Grundzuge  der  Differential  und  Integralrechnung,  p.  247)  ascribes  this 
mode  of  expression  to  A.  Mayer  (see  paper  by  him  in  the  Leipz.  Ber.  (1889),  p.  128). 
The  form  as  presented  here  is  found  in  Weierstrass's  lectures  delivered  at  least  ten 
years  before  the  Mayer  paper. 


80  THEORY  OF  MAXIMA  AND  MINIMA 

This  is  the  usual  form  of  Taylor's  theorem  for  functions  of  several 
variables.    In  particular,  when  m  =  1  the  above  development  is 
[5]     f(xv  x2,  .  .  .,  xn)  -/(ap  aa,  -  •  -,  an) 


H 


The  function  f(xv  •  •  .,  a?TO)  is  regular  and  continuous,  as  are 
consequently  all  its  derivatives.  If,  therefore,  the  first  deriva 
tives  of  f(xv  x2,  •  •  .,  xn)  are  all,  or  in  part,  ^0  for  xl=  alt  -  -  •, 
%n  =  an>  tnen  they  will  also  be  different  from  zero  for  x1  =  a1  +  0hly 
•  •  .,  xn=  an+  0hn,  where  the  absolute  values  of  hlt  &2,  •  •  •,  hn  have 
been  taken  sufficiently  small  ;  these  derivatives  will  also  be  of  the 
same  sign  as  they  were  for  x1  =  av  x2  =  a2,  .  .  .,xn=  an.  If,  now,  we 
choose  all  the  h's  zero  with  the  exception  of  one,  which  may  be 
taken  either  positive  or  negative,  it  is  seen  that  when  the  corre 
sponding  derivative  has  either  sign,  we  may  always  bring  it  about 
at  pleasure  that  the  difference 

f(xv  x2,  .  .  .,  xn)-f(av  a2)  .  .  .,  an) 

is  either  a  positive  or  a  negative  quantity,  and  consequently  at 
the  position  av  a2,  •  •  .,  an  no  extreme  value  of  the  function  is 
permissible. 

We  therefore  have  the  following  theorem  : 

Extremes  of  the  function  f(xv  x2,  •  •  .,  xn)  can  only  enter  for 
those  systems  of  values  of  (x1}  x2,  •  •  •,  xn)  which  at  the  same  time 
satisfy  the  n  equations  (p.  17) 

[6]  f  =0,     f  =0,     .-,     f  =  0. 

0x^  dx2  dxn 

It  may  happen  that  for  the  common  roots  of  the  system  of 
equations  [6]  still  higher  derivatives  also  vanish.  In  this  case 
we  can  in  general  only  say  that  if  for  a  system  of  roots  of  the 
equations  [6]  all  the  derivatives  of  several  of  the  next  higher 
orders  vanish,  and  if  the  first  derivative  which  does  not  vanish 
for  these  values  is  of  an  odd  order,  the  function,  as  may  be 
shown  by  a  method  of  reasoning  similar  to  that  above,  has 
certainly  no  maximum  or  minimum  value. 


NO  SUBSIDIARY  CONDITIONS  81 

52.  If,  however,  this  derivative  is  of  an  even  order,  then  in  the 
present  state  of  the  theory  of  forms  of  the  nth  order  in  several 
variables  there  is  no  general  criterion  regarding  the  behavior  of 
the  function  at  the  position  in  question.  We  therefore  limit  our 
selves  to  the  case  where  the  derivatives  of  the  second  order  of  the 
function  f(xv  x.2  ,  •  •  •  ,  xn)  do  not  all  vanish  for  the  system  of  real 
roots  alt  a2,  •  •  •,  an  of  the  equations  [6]. 

In  this  case  we  have  a  criterion  in  the  formula 

[7]     f(xlt  z,2,  .  .  .,  xn)  -f(av  a.2,  .  .  .,  aj 


by  which  we  may  determine  whether  f(xv  x2,  •  •  •,  xn)  has  an  ex 
treme  value  on  the  position  alt  «2,  •  •  •  ,  an,  since  we  may  determine 
whether  the  integral  homogeneous  function  of  the  second  degree, 


17 

l\ 


in  the  n  variables  hv  h2,  •  •  •,  hn  is  for  arbitrary  values  of  those 
variables  invariably  positive  or  invariably  negative. 

Denote  this  function  by  /AM(a1+  0hv  •••,#„+  6hn). 

On  account  of  their  presupposed  continuity  the  quantities 


/cy(xv  x2,...,  xn)\  and  /Pf(xlt  x2,  .  .  .,  xn)\ 

dxjx^  /«1+i»I,...,«.+ffc,  GX^  /a,,...,  an 

with  values  of  hlf  h^,---,  hn  taken  sufficiently  small  differ  from 
each  other  as  little  as  we  wish  and  are  of  the  same  sign  ;  *  hence 
with  small  values  of  the  h's  the  functions 


have  always  the  same  sign,  and  we  may  therefore  confine  our 
selves  to  the  investigation  of  the  latter  function. 

*  If  any  of  the  quantities  (  -  ,1*       '        )  becomes  zero,  we  may  replace 

\ 


it  by  exM(ai>  •  •  •  >  an)>  which  must  of  course  be  given  the  same  sign  as  /A,H(«I+  Ohi, 
an  +  0hn),  e\n  denoting  an  infinitesimally  small  quantity. 


82  THEORY  OF  MAXIMA  AND  MINIMA 

If  it  is  found  that  through  a  suitable  choice  of  hv  h2,  •  •  -,  kn 
the  expression 


can  be  made  at  pleasure  either  positive  or  negative,  the  same  will 
be  the  case  with  the  difference/^,  x2,  •  •  .,  xn)  —  f(alt  a2,  •  •  -,  a.w), 
and  consequently  f(xv  x2,  •  •  •,  xn)  has  on  the  position  (av  a2,  •  •  •  ,  an) 
no  extreme  value. 

We  therefore  have  as  a  second  condition  for  the  existence  of 
a  maximum  or  a  minimum  of  the  function  f(xlf  x2,  •  •  •  ,  xn)  on 
the  position  (av  a2,  •  >  •,  an)  that  in  case  the  second  derivatives 
of  the  function  f(xlt  x2,  •  •  •  ,  xn)  do  not  all  vanish  at  this  position, 
the  homogeneous  quadratic  form 


\ 


must  be  always  negative  or  always  positive  for  arbitrary  values 
of  hv  A2,  •  •  .,  hn. 

II.  THEORY  OF  THE  HOMOGENEOUS  QUADRATIC  FORMS 

53.  The  three  kinds  of  quadratic  forms,  viz.,  definite,  semi- 
definite,  and  indefinite,  were  denned  in  §  13. 

As  we  have  already  indicated  in  §  13,  it  is  seen  that  if  the  homo 
geneous  function  is  an  indefinite  form,  the  function  f(xv  x%,  >  •  •,  xn) 
has  neither  a  maximum  nor  a  minimum  upon  the  position  (av  a2, 
•  •  .,  aw);  for  if  the  right-hand  member  of  [7]  is  positive,  say,  for 
a  definite  system  of  values  of  the  h's,  then  in  accordance  with 
the  definition  of  the  indefinite  quadratic  forms  we  can  find  in 
the  immediate  neighborhood  of  the  first  system  a  second  system 
of  values  of  the  7^'s  for  which  the  right-hand  side  of  the  equa 
tion  [7]  is  negative  ;  consequently,  also,  the  difference 

f(xlt  a?2,  •  •  -,  xn)  -/(«!,  a2,  .  .  .,  an) 

is  negative,  so  that  therefore  no  maximum  or  minimum  is  permis 
sible  for  the  position  (alf  a2,  •  •  .,  a.n). 


NO  SUBSIDIARY  CONDITIONS 


83 


If,  then,  the  second  derivatives  of  the  function  f(xv  #2,  •  •  •,  xn) 
do  not  all  vanish  at  the  position  (av  «2,  •  •  •,  an),  it  follows,  besides 
the  equations  [6],  as  a  further  condition  for  the  existence  of  an 
extreme  of  the  function  f(xv  x2,  •  •  .,  x2)  that  the  terms  of  the 
second  dimension  in  [4]  must  be  a  definite  quadratic  form,  if  we 
exclude  what  we  have  called  the  semi-definite  case. 

The  question  next  arises :  Under  what  conditions  is  in  general 
a  homogeneous  quadratic  form 

[8]  +(0^...,.* 


a  definite  quadratic  form  ? 

54.  Before  we  endeavor  to  answer  this  question  we  must  yet 
consider  some  known  properties  of  the  homogeneous  functions  of 
the  second  degree. 

Suppose  that  in  the  function  (f)(xlf  x2,  •  -  •,  xn),  in  the  place  of 
(xv  x2,  •  -  •  ,  xn),  homogeneous  linear  functions  of  these  quantities 


[9] 


are  substituted,  which  are  subjected  to  the  condition  that 
inversely  the  x's  may  be  linearly  expressed  in  terms  of  the  £/'s, 
and  consequently  the  determinant 


[10] 


ll»  C12'  '  '  *'  C\n 
21'  C22'  "  '  "»  C2n 


The  function  </>(^,  «J2,  .  •  .,  a;n)  is  thereby  transformed  into 
[11]  <f>(xv  a?2,  •  •  -,  *w)=  -f  (yi,  y2,  .  .  .,  yn). 


Owing  to  this  substitution  it  may  happen  that 

does  not  contain  one  of  the  variables  y,  so  that  <f>(xv  x%, 

is  expressible  as  a  function  of  less  than  n  variables. 


,yn) 


84  THEORY  OF  MAXIMA  AND  MINIMA 

To  find  the  condition  for  this  write 


If  i/r  is  independent  of  one  of  the  y's,  say  yn,  so  that  conse 
quently  —  =  0,  then  from  the  n  equations 


we    may    eliminate    the    n  —  1    unknown    quantities    —  —  >  —  —  > 

dojr  ^    ^2 

•  •  •  ,  —  —  •    We  thus  have  among  the  <f>'s  an  equation  of  the  form 


[14] 


where  the  &'s  are  constants. 

Owing  to  equations  [12]  this  means  that  the  determinant  of 
the  given  quadratic  form  vanishes,  that  is, 


[15] 

We  note  here  the  following  formulas: 


[16] 


and  consequently 

[17] 


There  exists,  further,  the  well-known  Euler's  theorem  for  homo 
geneous  functions  : 

[I8] 

It  is  also  easy  to  show  reciprocally  that  if,  as  above,  the  equa 
tion  [15]  is  true,  the  function  <f>  consists  of  less  than  n  variables. 


NO  SUBSIDIARY  CONDITIONS  85 

For  if  we  assume  that  equation  [15],  or,  what  amounts  to  the 
same  tiling,  an  identical  relation  of  the  form  [14]  exists,  and  if 
we  substitute  in  4>(xly  #2,  •  •  •,  xn)  the  quantities  #A  -f-  tk^  in  the 
place  of  #A(\  =  1,  2,  •  •  •,  n)  and  develop  with  respect  to  powers 
of  t,  we  then  have 

<t>  (xl  +  tkv  x2  +  tk2,  •  •  •  ,  xn  +  tkn) 

=  $(xl9  a?2,  •  •  •,  xn)  +  2  t       {k^(x1}  x2,...,  xn)} 


It  follows,  when  we  take  into  consideration  the  equations  [14] 
and  [18],  since  the  equation  [14]  is  true  for  every  system  of 
values  (xv  x2,  •  •  •,  xn),  that 

<f>(xl  +  tkv  -  •  .,  xn  +  *&„)  =  <£  (a^,  •  •  .,  «„). 

Hence,  if  the  equation  [15]  exists,  or  if  the  k's  satisfy  the  equa 
tion  [14]  for  every  system  of  values  (xv  x2,  •  •  •,  xn),  then 
^to,  x2,  -  •  .,  xn)  remains  invariantive  if  x^+tkK  is  written  for 
#A,  where  t  is  an  arbitraiyr  quantity. 

Consequently,  it  being  presupposed  that  kv  ^  0,  if  t  is  so 
determined  that  the  argument  .«'„  -+•  ^  =  0,  we  have 

[19]     ^(jjj,  jL-2,  •  •  .,  ^^^Ui-^v,  x<t--^xv,  •  .  ., 

/i/y  _  -j  ^.  ft  „  _i_  •« 

xv-l         T      ^  ">  Xv  +  l          T       ^ 

where  $  is  expressed  as  a  function  of  less  than  n  variables. 
We  therefore  have,  the  theorem 

The  vanishing  of  the  determinant^?  ±AUA22  •••  Ann  is  the 
necessary  and  sufficient  condition  that  a  homogeneous  quadratic 
function  <t>(xv  x2,  •  •  •,  xn)=^A^xxx^  be  expressible  as  a  func 
tion  of  n  —  1  variables.  A'  ** 

55.  We  return  to  the  question  proposed  at  the  end  of  §  53, 
and  to  have  a  definite  case  before  us  assume  that  the  problem  is  : 
Determine  the  condition  under  which  the  function  <£(#j,  x%,  •  •  •,  xn) 
is  invariably  positive.  The  second  case  where  <f>(xly  x2,  •  •  •,  xn) 
is  to  be  invariably  negative  is  had  at  once  if  —  (f)  is  written  in 
the  place  of  $. 


*^ii          \ 

>  xn~  ~T  Xv\> 


86  THEORY  OF  MAXIMA  AND  MINIMA 

We  shall  first  show,  following  a  method  due  to  Weierstrass,* 
that  every  homogeneous  function  of  the  second  degree  <£(<#!,  «£2, 
.  .  .,  xn)  may  be  expressed  as  an  aggregate  of  squares  of  linear 
functions  of  the  variables. 

56.  In  the  proof  of  the  above  theorem  it  is  assumed  that 
(f>(xv  x2,  •  •  •,  xn)  cannot  be  expressed  as  a  function  of  n—1 
variables  ;  it  follows,  therefore,  that  the  inequality 

[20]  ^±AuA22...Ann^O 

is  true  and  that  therefore  it  is  not  possible  to  determine  con- 

i  =  ")i 

stants  k,  so  that  the  equation  ^ki(f>i  =  0  exists  identically. 

i  =  l 

If,  then,  y  is  a  linear  function  of  the  x's  having  the  form 
[21]  y  =  C&  +  c2^2  H  ----  -f-  cnxn, 


and  if  g  is  a  certain  constant,  then  the  expression  <$>—  gy*  (=<$>, 
say),  after  the  theorem  proved  above,  can  be  expressed  as  a 
function  of  only  n  —  1  variables  if  the  constants  kv  k%,  •  •  •,  kn 
may  be  so  determined  that 


or 

[22" 


A=l 


From  the  assumption  made  regarding  [20]  it  follows,  on  the 
one  hand,  that  the  inequality 

[23]  2)^x^0 

A 

must  exist.    This  is  the  only  restriction  placed  upon  the  c's.    On 
the  other  hand,  in  virtue  of  the  n  linear  equations 

[24]  2>W=&      (X=l,  2,.-.,») 

ft 

*  See  also  Lagrange,  Misc.  Taur.,  Vol.  I  (1759),  p.  18,  and  Mtcanique,  Vol.  I,  p.  3; 
Gauss,  Disq.  Arithm.,  p.  271  ;   Theoria  Comb.  Observ.,p.  31,  etc. 


NO  SUBSIDIAKY  CONDITIONS  87 

the  quantities  xv  #a,  •  •  •,  xn  may  be  expressed  as  linear  functions 
of  <f>v  <£2,  •  -  •,  <£„,  and,  consequently,  by  the  substitution  of  these 
values  of  xv  x2,  -  -  -,  xn  in  [21]  y  takes  the  form 

V=  It 

[25]  y 

where  the  ev  are  constants,  which  are  composed  of  the  constants 
A^  and  CA. 

But  from  equation  [22]  it  follows  that 


Such  a  representation  of  the  $A,  however,  since  we  have  to  do 
with  linear  equations,  can  be  effected  only  in  one  way. 

l  2)*A    *.« 

We  therefore  have  y  =  --^  -  =$)e*^ 

"SU  A=1 

M=I 
from  which  it  follows  that 

fe-^S1**     (X=*l,2,...,»). 

^=1 

Through  the  substitution  of  these  values  in  [22]  it  is  seen  that 

X  =  M  A  =  it 

5X*A-0y5X«x=o; 

A=l  A=l 

consequently,  owing  to  the  relation  [25],  we  have 
[26]  =  t«» 


This  value  of  g  may  be  expressed  in  a  different  form  ;  for  from 
[25]  and  [17]  it  follows  that 

V  =  H  V  *=  71 

[26a]  y  =5)«A(a;i»  «a,  •  •  •,  xn) 


88  THEORY  OF  MAXIMA  AND  MINIMA 

Comparing  this  result  with  [21],  we  have 

[27]  cv=  <£„(«!»  «2,  •  •  •,  en)      (v  =  1,  2,  •  •  .,  TI), 

and  consequently 


or,  from  [18], 

«!,«,,  •••>'„ 

Since  the  quantities  cp  c2,  •  •  •,  cw  are  perfectly  arbitrary  except 
the  one  restriction  expressed  by  the  inequality  [23],  the  quantities 
elt  «2>  •  •  •>  en  are»  in  consequence  of  the  equation  [27],  completely 
arbitrary  with  the  one  limitation  resulting  from  [28],  viz.,  the 
function  <p  cannot  vanish  for  the  system  of  values  (ev  e%,  •  •  •,  en)- 
otherwise  g  would  become  infinite. 

57.  Reciprocally,  if  the  quantities  ev  e2,  •  •  •,  en  are  arbitrarily 
chosen,  but  with  the  restriction  just  mentioned,  and  if  g  is 
determined  through  [28],  it  may  be  proved  that  the  expression 
(f)(=  $>  —  gy*\  where  y  has  the  form  [25],  may  be  expressed  as 
a  function  of  only  n  —  I  variables.  For,  form  the  derivatives  of 
this  expression  with  respect  to  the  different  variables,  and  multiply 
each  of  the  resulting  quantities  by  the  constants  ev  ez,  -  •  -,  en. 
Adding  these  products  and  noting  [26a]  and  [28],  we  have 


The  expression  on  the  right-hand  side  is  zero  from  [25].  Hence 
n  constants  may  be  chosen  in  such  a  way  that  the  sum  of  the 
products  of  these  constants  and  the  derivatives  of  the  expression 
(f>  —  gy2  is  identically  zero,  and  also  ^(elt  •  •  •,  en)  =  0  (cf.  [18]). 
58.  Substitute  xx+tex  for  a3A(\  =  l,  2,  •  •  .,  n)  in  <f)  ;  if  one  of 
these  arguments  is  made  equal  to  zero,  we  have,  as  in  §  54, 


ek  ek  ek  ek 

or,  if  the  new  arguments  are  represented  by  x'v  x'2)  •  •  •,  x'n_v 
<f>(xv  x2,  .  .  .,  xn)  -  gy2  =  $(x'v  x'2>  -  •  -,  a/n_j). 


NO  SUBSIDIARY  CONDITIONS  89 

Employing  the  same  method  of  procedure  with  </>  (x'v  x'2,  •  •  ',x'n  _x) 
as  was  done  with  <f>  (xv  #2,  •  •  •,  xn),  we  come  finally  to  the  func 
tion  of  only  one  variable,  which,  being  a  homogeneous  function  of 
the  second  degree,  is  itself  a  square.  Hence  we  have  the  given 
homogeneous  function  <$>(xv  x2,  •  •  •,  xn)  expressed  as  the  sum  of 
squares  of  linear  homogeneous  functions  of  the  variables.  If  the 
coefficients  of  <f>  are  real,  as  also -the  quantities  e,  the  coefficients  g 
are  also  real,  and  since  the  quantities  e  may  with  a  single  limi 
tation  be  arbitrarily  chosen,  it  follows  that  a  transformation  of 
such  a  kind  that  the  result  shall  be  a  real  one  may  be  performed 
in  an  infinite  number  of  ways.* 

59.  If,  now,  the  expression 

[29]         </>  (xlt  ffa,  •  •  • ,  xn)  =  gfl*  +  g$l  H +  gnyl 

is  to  be  invariably  positive  for  real  values  of  the  variables  and 
equal  to  zero  only  when  the  variables  themselves  all  vanish,  then 
all  the  qualities  glt  g2,  •  •  -,  gn  must  be  positive;  for  if  this  were 
not  the  case,  but  gv  say,  were  negative,  then,  since  the  y's  are, 
independently  of  one  another,  linear  homogeneous  functions  of  the 
#'s,  we  could  so  choose  the  x's  that  all  the  ?/'s  except  yl  would 
vanish,  and  consequently,  contrary  to  our  assumption,  <t>(xv  x2, 
•  •  • ,  xn)  would  be  negative.  Furthermore,  none  of  the  gs  can  vanish ; 
for  if  (/j,  say,  were  zero,  we  might  so  choose  a  system  of  values 
xv  x2, . .  .,  xn,  in  which  at  least  not  all  the  quantities  xv  #2,  •  •  .,  xn 
were  zero,  that  all  the  y's  would  vanish  except  yv  and  consequently 
<f>  could  then  be  zero  without  the  vanishing  of  all  the  variables 

•^l>  ^2'  *  *  *'  ^n* 

Eeciprocally,  the  condition  of  gv  g2,  •  •  •,  gn  being  all  positive  is 
also  sufficient  for  </>  to  be  invariably  positive  for  real  values  of 
the  variables,  and  for  <f>  to  be  equal  to  zero  only  when  all  the 
variables  vanish. 

60.  In  order  to  have,  in  as  definite  form  as  possible,  the  ex 
pression  of  $  as  a  sum  of  squares,  we  shall  give  to  the  expression 
[26]  for  g  still  a  third  form. 

*See  Burnside  and  Panton,  Theory  of  Equations  (1892),  p.  430.  In  this  connec 
tion  it  is  of  interest  to  note  the  Theorem  of  Inertia  of  Sylvester,  Coll  Math.  Papers, 
Vol.  I,  pp.  380,  511.  See  also  Hermite,  CEuvres,  Vol.  I,  p.  429. 


90  THEORY  OF  MAXIMA  AND  MINIMA 

In  connection  with  [12]  it  follows  from  [27]  that 

°v=  ^  A^M     (v  =  1,  2, .  •  .,  n). 
M=I 

Denote  by  A  the  determinant  of  these  equations,  which  from  [20] 
is  not  identically  zero,  that  is, 


[30]  ^ 

We  have  as  the  solution  of  the  preceding  equation 


It  follows  from  this  in  connection  with  [26]  that 
[31]  9  = ^ 


an  expression  in  which  the  c's  are  subject  only  to  the  one  con 
dition  that 


is  not  identically  zero. 

61.  It   is   shown   next   that   we   may  separate   from  <f>(x^  x2, 

•  •  •,  xn)  the  square  of  a  single  variable  in  such  a  way  that  the 
resulting  function  contains  only  n  —  1  variables. 

For  example,  in  order  that  the  expression  (f>  —  gx%  be  expressed 
as  a  function  of  n  —  1  variables,  we  may  choose  for  g  the  value 
[31],  after  we  have  written  in  this  expression  cx=0(X  =  l,  2, 

•  •  •,  n  —  1),  while  to  ctl  is  given  the  value  unity. 

From  this  it  is  seen  that 


A      _A_ 

3A    ~A 


where  A1  is  the  determinant  of  the  quadratic  form  $(xv  x2,  .  .  ., 
#»_!>  0).    Of  course  this  determinant  must  be  different  from  zero. 


XO  SUBSIDIARY  CONDITIONS  91 

Hence  we  may  write 

<t>(xv  xv  -  -  .,  &«)  =  -r-a£+$(3i,  4>  '  *  '»  ^'-i)> 
<*! 

where 

-  ~  --          '  '  '  ~  ^ 


We  may  then  proceed  with  </>  just  as  has  been  done  with  <f>  by 
separating  the  square  of  x^_lf  etc. 

After  the  separation  of  /-t  squares  from  the  original  function  <f>, 
we  notice  that  the  determinant  of  the  resulting  function  in  n  —  /z 
variables  is  the  same  as  the  determinant  of  the  function  which 
results  from  the  original  function  (/>  when  wre  cause  the  JJL  last 
variables  in  it  to  vanish.  If  this  determinant  is  denoted  by  A^ 
we  have  the  following  expression  for  </>  : 

2,  •  .  .,  xn) 


-  1 


62.  If  now  <t>  is  to  be  invariably  positive  and  equal  to  zero  only 
when  all  the  variables  vanish,  the  coefficients  on  the  right-hand 
side  of  the  above  expression  must  all  be  greater  than  zero.  We 
therefore  have  the  theorem 

In  order  that  the  quadratic  form 


be  a  definite  form  and  remain  invariably  positive,  it  is  necessary 
and  sufficient  that  the  quantities  Av  AZ)'->,An_v  which  are 
defined  through  the  equation  A^  =  ^±AUA^'  •  -A,^^^,,^,  be  all 
positive  and  different  from  zero.  If,  on  the  other  hand,  the  quad 
ratic  form  is  to  remain  invariably  negative,  then  of  the  quantities 
An_v  An_2,  •  •  •,  Alt  A,  the  first  must  be  negative,  and  the  following 
must  be  alternately  positive  and  negative  (see  Stolz,  Wiener  Bericht, 
Vol.  LVIII  (1868),  p.  1069). 


92  THEORY  OF  MAXIMA  AND  MINIMA 

III.    APPLICATION    OF   THE    THEORY   OF   QUADRATIC 

FORMS  TO  THE  PROBLEM  OF  MAXIMA  AND  MINIMA 

STATED  IN   §§47-51 

63.  By  establishing  the  criterion  of  the  previous  section  the 
original  investigation  regarding  the  maxima  and  minima  of  the 
f  unction  f(xlt  x2,  •  -  •,  xn)  is  finished.  The  result  established  in  §  57 
may  in  accordance  with  the  definitions  given  in  §  52  be  expressed  as 
follows:  In  order  that  an  extreme  of  the  function  f(xv  x2,  •  •  •  ,  xn) 
may  in  reality  enter  on  the  position  (av  a2,  •  •  •,  an)  which  is  deter 
mined  through  the  equations  [6],  it  is  sufficient,  if  the  second 
derivatives  of  the  function  do  not  all  vanish  at  this  position,  that 
the  aggregate  of  the  terms  of  the  second  degree  of  the  equation  [4] 
be  a  definite  *  quadratic  form  ;  if,  however,  the  form  vanishes  for 
other  values  of  the  variables  without  changing  sign  (that  is,  is  semi- 
definite),  then  a  determination  as  to  whether  an  extreme  in  reality 
exists  is  not  effected  in  the  manner  indicated  and  requires  further 
investigation,  as  is  seen  below. 

In  virtue  of  the  theorem  stated  in  §  53,  an  extreme  will  enter 
for  a  system  of  real  values  of  the  equation  [6]  if  the  homogeneous 
function  of  the  second  degree 


that  is,  if 


is  a  definite  quadratic  form  ;  in  other  words  (§  62),  there  will  be  a 
minimum  on  the  position  (av  a2,  -  •  •,  an)  if  the  quotients 


where  ^M  =  ^  ±/1i/22+  •  •  •  A-,.,™-M>  are  al1  positive,  a  maxi 
mum  if  they  are  all  negative.  In  both  cases  the  quotients  must  be 
different  from  zero. 

*Lagrange,  Thtorie  des  Fonctions,  pp.  283,  286;   see  also  Cauchy,  Gale,  differ., 
p.  234. 


NO  SUBSIDIARY  CONDITIONS  93 

This  last  condition  is  only  another  form  of  what  was  said 
above,  viz.,  that  2J/A*W^  mus^  n°t  be  a  semi-definite  form. 
For  if,  say,  A'  * 


then  the  summation  ^f^hji^  being  denoted  by  (f>(JivhZ)  •  •  •,&„), 

I.* 
this  equation  would  directly  imply  the  existence  of  a  relation  of 

the  form 


where  the  £„  are  constants  which  do  not  all  simultaneously  vanish. 
If,  therefore,  knt  say,  is  different  from  zero,  we  may  write 


and  with  the  help  of  this  relation  we  have  from  the  equation 


A=l  X=l 

the  following  relation 


Now  in  this  expression  the  arbitrary  quantities  li  may  be  so 
chosen  that  ^ 

ht  =  -±hn     (X=],  2,...  ,TI), 
*« 

and  consequently  the  function  0(7^,  7i2,  •  •  .,  AH)  would  vanish 
without  all  the  tis  becoming  simultaneously  zero.  This  case  we 
cannot  treat  in  its  generality. 

Neglecting  this  case,  it  is  seen  that  the  problem  of  this  chapter 
is  completely  treated  ;  however,  the  conditions  that  a  quadratic 
form  shall  be  a  definite  one  appear  in  a  less  symmetric  form 
than  we  wish.  It  is  due  to  the  fact  that  we  have  given  special 
preponderance  to  certain  variables  over  the  others. 

We  shall  consequently  take  up  the  same  subject  again  in  the 
next  chapter. 


94  THEOHY  OF  MAXIMA  AND  MINIMA 

64.  The  question  is  often  regarding  the  greatest  and  the  least 
values  (the  upper  and  lower  limits)  which  a  function  may  take 
when  its  variables  vary  in  a  given  finite  or  infinite  region.  If 
this  value  corresponds  to  a  system  of  values  within  the  given 
region,  then  for  this  system  the  function  will  also  be  a  maximum 
or  a  minimum  in  the  sense  derived  above. 

For  example,  let  it  be  required  to  distribute  a  positive  number 
a  into  n  +  l  summands,  so  that  the  product  of  the  ^th  power 
of  the  first,  the  «2th  power  of  the  second,  etc.,  and  finally  the 
an  +  i  P°wer  °f  the  last  summand  will  be  a  maximum.* 

The  quantities  av  a2,  •  .  .,  an  +  l  are  to  be  positive  numbers. 
Let  xv  x2,  •  •  .,  xn,  a  —  xl  —  x2  —  .  .  .  —  xn  be  the  summands  in 
question  and  write 

U=  x^xf*  -  -  -  xna«(a  -  Xl -  x2 O""*1. 

We  must  then  determine  when  U  or,  what  is  the  same  thing, 
its  natural  logarithm,  has  its  greatest  value. 

If  we  put  the  partial  derivatives  of  log  U  equal  to  zero,  we*  will 


-o, 


These  equations  may  be  written 


the   last    term    being    had    through    addition    of    the    preceding 
proportions. 

If  we  caH  x®>,  x^\  .  .  .,  ^0)  the  values  of  the  variables  which 
satisfy  these  equations,  we  have 

a, 


*  Peano,  §  137. 


NO  SUBSIDIARY  CONDITIONS  95 

The.  corresponding  value  of  U  is 


To  recognize  whether  U0  is  the  greatest  of  the  values  of  U,  we 
may  show  that  U  is  in  fact  a  maximum  for  the  system  of  values 
xf\  .  .  .,  x$  and  that  this  position  lies  on  the  interior  of  the  realm 
of  variability  under  consideration.  For,  let  xv  x2,  •  •  -,  xn  be  an 
other  system  of  positive  values  of  the  variables,  for  which  also 
a  —  x1  —  ...  —  xn  is  positive,  and  substitute  for  the  variables  in 
log  U  the  values 

af>  +  u  (xl  -  af>),  .  .  •  ,  x®  +  w  (*tt  -  zf),     where     0  <  u  <  1. 

Since  the  partial  derivatives  of  the  first  and  second  order  of  log  U 
are  continuous  for  all  these  systems  of  values,  we  have  through 
the  Taylor  development,  observing  that  the  first  derivatives  vanish 
on  the  position  x^\  •  •  •,  x® 


, 


1  \rr  fr®        r( 

-\ovV  --  \    l(  l    ~    l 
0     2[       (^))2 


(«-«P  -----  41})2 

where  icj15,  •  •  •  ,  x$  are  values  of  the  variables  of  the  form 
^+0xl-xf...,x(®+6xn-x(®,    where    0<0<1. 


The  expression  within  the  brackets  is  positive  and  different 
from  zero,  since  it  is  assumed  that  the  system  of  values  xly  #2, 
.  .  .  ,  xn  do  not  coincide  with  x®>  ,  x^  ,  «  •  «  ,  x®  .  It  follows  that 
log  U  <  log  UQ  or  U  <  UQ,  so  that  Z70  is,  in  fact,  the  greatest 
value  which  U  can  assume. 

We  note  that  U  takes  a  smallest  value,  viz.,  zero,  if  one  of  the 
summands  into  which  a  is  distributed,  vanishes.  If  we  allow  the 
summands  to  take  negative  values,  it  no  longer  follows  that  UQ 
is  the  greatest  of  the  values  U. 


CHAPTER  VI 

THEORY  OF  MAXIMA  AND  MINIMA  OF  FUNCTIONS  OF  SEVERAL 

VARIABLES  THAT  ARE  SUBJECTED  TO  SUBSIDIARY  CONDITIONS. 

RELATIVE  MAXIMA  AND  MINIMA 

65.  In  the  preceding  investigations  the  variables  xv  x2,  •  -  .,  xn 
were  completely  independent  of  one  another. 

We  now  propose  the  problem :  Among  all  systems  of  values 
(xv  x2,  •  •  .,  xn)  find  those  which  cause  the  function  F(xl>  %%,•••,  xn) 
to  have  maximum  and  minimum  values  and  which  at  the  same 
time  satisfy  the  equations  of  conditions : 

[I]      f^(x1}  x2,  •  ••,  O=0      (X  =  l,  2,  .  .  .,  m-  m<n), 

where  f^(xv  x2,  •  •  .,  xn)   and  F(XI}  •  •  •,  xn)   are  functions  of  ike 
same  character  as  f(xlf  #2,  •  •  •  ,  xn)  in  §  47. 

66.  The  natural  way  to  solve  the  problem  is  to  express  by  means 
of  equations   [1]  m  of  the  variables  in  terms  of  the  remaining 
n  —  m  variables  and  write  their  values  in  F(XI}  x%,  •  •  •,  xn).    This 
function  would  then  depend  only  upon  the  n  —  m  variables  which 
are  independent    of    one   another,   and   so    the    present    problem 
would  be  reduced  to  the  one  of  the  preceding  chapter. 

In  general,  this  method  of  procedure  cannot  be  readily  per 
formed,  since  it  is  not  always  possible  by  means  of  equations  [1] 
to  represent  in  reality  m  variables  as  functions  of  the  n  —  m  remain 
ing  variables.  A  more  practicable  method  must  therefore  be  sought. 

67.  If  (av  a2,  •  •  -,  an)  is  any  system  of  values  of  the  quantities 
xi>  x%> '  '  '>  xn  which  satisfy  .the  equations  [1],  then  of  the  systems 

of  values 

(Xl  =  a1  +  fcj,  &,=  aa-f  A,,  •  •  • ,  xn=  an  +  hn), 

in  the   neighborhood   of   (av  •  •  •,  an),   only  those   which   satisfy 
the  equations   [1]  may  be  considered;  that  is,  we  must  have 

[2]   /A(a1+^1,a2+^2,.",an+^w)=0     (X  =  1,  2,  .  .  .,  m). 


RELATIVE  MAXIMA  AND  MINIMA  97 

Hence  by  Taylor's  theorem  the  h's  satisfy  the  equations 

[3]  {/AM  («1»  a*>  '  '  •>  an)  U  +  [hlt  h2,  •  •  .,  hn]  I  =  0 


where  [Aj,  h2,  •  •  >,  hn]j*  denotes  the  terms  of  the  second  and  higher 
dimensions  in  the  respective  variables. 

68.  It  being  assumed  that  at  least  one  of  the  determinants 
of  the  rath  order  which  can  be  produced  by  neglecting  n  —  m 
columns  from  the  system  of  m  •  n  quantities 


[4] 


/Ml  J 22'  *  *  *J  /2n> 


_/ml>«//n2>  *  '  ''Jmn* 

is  different  from  zero,  then  (see  §§135  and  136)  m  of  the  quan 
tities  h  may  be  expressed  through  the  remaining  n  —  m  quantities 
(which  may  be  denoted  by  klf  kz,  •  •  .,  kn_m)  in  the  form  of  power 
series  as  follows  : 

[5]   h,=  (kl,  A-2,  .  .  .,  *„_  J 


where  the  upper  indices  denote  the  dimensions  of  the  terms  with 
which  they  are  associated.  These  series  converge  in  the  manner 
indicated  in  §  136  ;  they  satisfy  identically  the  equations  [2]  and 
furnish,  if  the  quantities  kv  k2,  •  •  .,  kn_m  are  taken  sufficiently 
small,  all  values  of  the  m  quantities  h  which  satisfy  these 
equations. 

69.  The  condition  that  one  of  the  determinants  in  the  preced 
ing  article  be  different  from  zero  is  in  general  satisfied;  there 
are,  however,  special  cases  where  this  is  not  the  case.  A  geo 
metrical  interpretation  will  explain  these  exceptions. 

Let  F  and  an  equation  of  condition  f  =  0  contain  only  three 
variables  xlt  x2,  and  #3. 

The  equation  of  condition  f(xl}  #2>  xz)  —  ®  represents  then  a 
surface  upon  which  the  point  (xl}  #2,  xs)  is  to  lie  and  for  which 
F(xv  x2,  x3)  is  to  have  a  maximum  or  minimum  value. 


98  THEORY  OF  MAXIMA  AND  MINIMA 

The  determinants  of  the  first  order  in  the  development 


with  respect  to  powers  of  7^,  h2,  and  hs  cannot  all  be  equal  to  zero; 
that  is,  all  the  terms  of  the  first  dimension  cannot  vanish,  the 
single  terms  being  these  determinants ;  and  this  means  that  the 
surface  /  =  0  cannot  have  a  singularity  at  the  point  in  question. 

Take  next  two  equations  of  condition  /:  =  0  and  /2  =  0  between 
three  variables  xv  x2,  and  XB.  Considered  together  they  represent 
a  curve,  and  the  condition  that  the  corresponding  determinants  of 
the  second  order  cannot  all  be  zero  means  here  that  the  curve  at 
the  point  in  question  cannot  have  a  singularity. 

70.  If  the  values  of  the  m  quantities  k^  are  substituted  in  the 
difference  ^/  x  x  \_p/a  a  a  \ 

this  expression  then  depends  only  upon  the  n  —  m  variables 
&i>  &2>  '  '  *>  kn-m>  that  are  independent  of  one  another  and  may 
consequently  for  sufficiently  small  values  of  these  variables  be 
developed  in  the  form 

[6]  F(xv  x2,  •  •  -,  xn)  —  F(alt  «2,  •  •  •,  an) 

P=n-m  -I 


It  was  seen  (§  51)  that,  in  order  to  have  a  maximum  or  minimum 
on  the  position  (av  a2,  -  -  .,  an),  it  is  necessary  that  the  terms  of 
the  first  dimension  vanish,  and  consequently 

[7]  <7p=0      (p=l,  2,  ...,n-m). 

71.  This  condition  may  be  easily  expressed  in  another  manner. 
We  may  obtain  the  quantities  e  if,  in  the  development 

F(xv  aja,  •  •  -,  xn)-F(av  a2,...,an) 


we  substitute  in  the  terms  of  the  first  dimension  the  values  of  the 
m  quantities  from  [5]  and  arrange  the  result  according  to  the 


RELATIVE  MAXIMA  AND  MINIMA  99 

quantities  kv  k2,  •  •  .,  kn_m.    In  other  words,  the  equations  [7]  ex- 

tL-  n 

press  the  condition  that  ^FJi^  must  vanish  identically  for  all 

*-i 
systems  of  values  of  the  h's  that  satisfy  the  m  equations  [3]  after 

they  have   been   reduced   to   their  linear  terms.    These  are  the 
m  equations 

"=o  *=i>  2,  •...». 


Now  multiplying  these  m  equations  *  respectively  by  m  arbitrary 
quantities  ev  e2,  -  -  .,  em,  and  adding  the  results  to  the  equation 


we  have  the  following  equation  : 

[9]  *2f{(^+  'I/I,  +  «2/2M  +   '  •  •  +  *mfmj*>3=  0. 

M  =  l 

But  the  es  may  be  so  determined  that  those  terms  in  this  sum 
mation  drop  out  which  contain  the  m  quantities  h,  which  are  ex 
pressed  in  [5]  through  the  n  —  m  other  h's  ;  by  causing  these 
terms  to  vanish,  a  system  of  m  linear  equations  is  obtained,  whose 
determinant  by  hypothesis  is  different  from  zero. 

Since  the  terms  which  remain  of  equation  [9]  are  multiplied 
by  the  completely  arbitrary  quantities  kv  k2,  •  •  .,  £„_,„,  it  is  not 
possible  for  this  equation  to  exist  unless  each  of  the  single 
coefficients  is  equal  to  zero. 

Consequently  we  have  as  the  first  necessary  condition  for 
the  appearance  of  a  maximum  or  minimum  the  existence  of 
the  following  system  of  n  equations, 


in  the  sense  that  if  m  of  these  equations  exist  independently  of 
one  another,  the  remaining  n  —  m  of  them  must  be  identically 
satisfied  through  the  substitution  of  the  e's  which  are  derived 

*This  method  is  due  to  Lagrange,  Theorie  des  Fonctions,  p.  268;  see  also  Gauss 
(Theoria  Comb.  Observ.   Supp.  §11). 


100  THEOKY  OF  MAXIMA  AND  MINIMA 

from  the  first  m  equation,  it  being  of  course  presupposed  that 
the  system  of  values  (alt  a2,  •  •  .,  an)  has  already  been  so  chosen 
that  the  equations  [1]  are  satisfied. 

Taking  everything  into  consideration  we  may  say :  In  order 
that  the  function  F  (xv  x2,  -  •  >,  xn)  have  a  maximum  or  mini 
mum  on  any  position  (av  a%,  •  •  •,  an),  it  is  necessary  that  the 
n  +  m  equations 


(/i  =  l,   2,-.. ,71), 

be  satisfied  by  a  system  of  real  values  of  the  n  +  m  quantities 

72.  These  deductions  were  made  under  the  one  assumption 
that  at  least  one  of  the  determinants  of  the  rath  order  which 
can  be  formed  out  of  the  m  •  n  quantities  [4]  through  the  omission 
of  n  —  m  columns  does  not  vanish.  This  condition  was  necessary 
both  for  the  determination  of  the  quantities  h,  which  satisfy  the 
equations  [2],  and  also  for  the  determination  of  the  m  factors  ev 

It  may  happen*  that  a  maximum  or  minimum  of  the  function 
F  enters  on  the  position  (av  «2,  •  •  .,  an)  even  when  the  above 
condition  is  not  satisfied.  For  if  it  is  possible  in  any  way  to 
determine  all  systems  of  values  of  the  h's  not  exceeding  certain 
limits  that  satisfy  the  equations  [2],  the  equations  [7]  together 
with  the  equations  [1]  are  sufficient  in  number  to  determine  the 
n  quantities  av  a2,  •  -  •,  an. 

When  the  above  assumption  is  not  satisfied,  the  equations  [8] 
exist  identically,  and  consequently  the  equations  [3],  which  serve 
to  determine  the  A's,  begin  with  terms  of  the  second  dimension. 
We  may  often  in  this  case  proceed  advantageously  by  introducing 
in  the  place  of  the  original  variables  a  system  of  n  —  m  new  vari 
ables  so  chosen  that  when  they  are  substituted  in  the  given 
equations  of  condition  they  identically  satisfy  them. 

*  See  Stolz,  p.  257. 


RELATIVE  MAXIMA  AND  MINIMA  101 

73.  To  make  clear  what  has  been  said,  the  following  example 
will  be  of  service  ;  its  general  solution  is  given  in  the  sequel  (§  91). 
Find  the  shortest  line  which  can  be  drawn  from  a  given  point  to 
a  given  surface.  Upon  the  surface  there  are  certain  points  of 
such  a  nature  that  the  lines  joining  these  points  with  the  given 
point  have  the  desired  property  and,  besides,  stand  normal  to  the 
surface  at  these  points. 

If  by  chance  it  happens  that  one  of  these  points  is  a  double 
point  (node)  of  the  surface,  so  that  at  it  we  have  fl  =  0,  /2  =  0, 
f3  =  0,  then  in  reality  for  this  point  the  terms  of  the  first  dimen 
sion  in  the  equations  [2]  drop  out  and  we  have  the  case  just 
mentioned. 

If  the  surface  is  the  right  cone 


we  may  write 


The  equation  of  the  surface  is  identically  satisfied,  and  it  is  easily 
seen  that  we  may  express  the  quantities  hlf  h2,  h3  through  two 
quantities  l\  and  &2  independent  of  each  other  even  in  the  case 
where  the  required  point  of  the  surface  is  the  vertex  of  the  cone, 
that  is,  the  point  x=  Q  =  y  =  z,  OT  u  =  0  =  v-  and  in  fact  in  such 
a  way  that  not  only  indefinitely  small  values  of  hlt  h2,  hs  corre 
spond  to  indefinitely  small  values  of  klf  A'2  but  also  that  all 
systems  of  values  hv  h%,  h3  are  had  which  satisfy  the  equation 


The  variables,  however,  must  be  given  at  one  time  real,  at  another 
time  purely  imaginary,  values  if  the  equations  [11]  are  to  repre 
sent  the  entire  surface  of  the  cone  ;  but  in  this  manner  the 
unavoidable  trouble  has  taken  such  a  direction  that  the  proposed 
problem  falls  into  two  similar  parts,  which  may  be  treated  in  full 
after  the  methods  of  Chapter  V.  In  other  cases  we  may  proceed 
in  a  like  manner.  The  special  problem  will  each  time  of  itself 
offer  the  most  propitious  method  of  procedure. 


102  THEORY  OF  MAXIMA  AND  MINIMA 

74.  We  must  now  establish  the  criteria  from  which  one  can 
determine  whether  a  maximum  or  minimum  oiF(x^  x2,-  -  •  ,  xu) 
really  enters  or  not  on  a  definite  position  (a1?  «2,  •  •  .,  a.n),  which 
has  been  determined  in  §  71  above. 

One  might  consider  this  superfluous,  since  in  virtue  of  the  cri 
teria  given  in  the  previous  chapter  a  maximum  or  minimum  will 
certainly  enter  if  the  aggregate  of  terms  of  the  second  dimension 
in  [6]  is  a  definite  quadratic  form  of  the  nature  indicated. 

It  is,  however,  desirable  to  determine  the  existence  of  a  maxi 
mum  or  minimum  without  having  previously  made  the  develop 
ment  of  the  function  in  the  form  [6];  for  in  order  to  obtain  the 
coefficients  Cpa.  we  must  pay  attention  not  only  to  the  terms  of 
the  first  dimension  but  also  to  the  terms  of  the  second  dimension, 
when  the  values  of  [5]  are  substituted  in  the  development  of 

F(xly  x2,  •  -  .,  xn)  —  F(av  «2,  • 


75.  The  above  difficulty  may  be  avoided  if  we  multiply  by 
the  quantities  elt(fji  =  1,  2,  •  •  •  m)  respectively  each  of  the  expres 
sions  [2]  which  vanish  identically,  add  them  thus  multiplied  to 
the  above  difference,  and  then  develop  the  whole  expression  with 
respect  to  the  powers  of  h. 

Owing  to  equation  [9]  terms  of  the  first  dimension  can  no 
longer  appear  in  this  development,  and  we  have,  if  we  write 

ft  =  m 

[12] 

[13]  F(xv  *2, .  .  . ,  xn)  -  F(alf  a»'...,  an)  =  G  (xv  x2, .  .  . ,  xn) 

1  v 

We  have,  accordingly,  the  homogeneous  function  of  the  second 
degree  ^Ofjcje^  of  the  formula  [6]  if  we  substitute  in  ^Gr^kph, 

p,cr  (JL,V 

the  values  [5]  and  consider  only  the  terms  of  the  first  dimension 


RELATIVE  MAXIMA  AND  MINIMA  103 

in  the  process.  If  then  the  criteria  of  the  preceding  chapter  are 
applied  we  can  determine  whether  the  function  F  possesses  or 
not  a  maximum  or  minimum  on  the  position  (av  a2,  .  .  .,  an). 

76.  The  definite   conditions  that  have  been  thus  derived  are 
unsymmetric  for  a  twofold  reason  :  on  the  one  hand  because  in 
the  determination  of  the  quantities  h  some  of  them  have  been 
given  preference  over  the  others,  and  on  the  other  hand  because 
those  expressions  by  means  of  which  it  is  to  be  decided  whether 
the  function  of  the  second  degree  is  continuously  positive  or  con 
tinuously  negative  have  been  formed  in  an  unsymmetric  manner 
from  the  coefficients  of  the  function. 

It  is  therefore  interesting  to  derive  a  criterion  which  is  free 
from  these  faults  and  which  also  indicates  in  many  cases  how 
the  results  will  turn  out.  With  this  in  view  let  us  return  to  the 
problem  already  treated  in  the  preceding  chapter  and  propose 
the  following  more  general  theorem  in  quadratic  forms. 

I.  THEORY  OF  HOMOGENEOUS  QUADRATIC  FORMS 

77.  THEOREM.     We  have  given  a  homogeneous  function  of  the 
second  degree 

[14]  <t>(xv  a  g,  •  •  .,  aw)=2)w4**2*a>      (A^  =  A^ 

A,M 

in  n  variables,  which  are  subjected  to  the  linear  homogeneous  equa 
tions  of  condition 


[15]        0A  =  2>AM.i'M=0     (X  =  l,  2,...,m;  m<n)-t 

we  are  required  to  find  the  conditions  under  which  <f>  is  invariably 
positive  or  invariably  negative  for  all  those  systems  of  values  of  the 
variables  which  satisfy  equations  [15]. 

It  is  in  every  respect  sufficient  to  solve  this  theorem  with 
the  limitation  that  the  quantities  x  are  subjected  to  the  further 
condition 

[16]  **  +  .**+...  +**  =  !; 

for  if  if  +  4  +  ...+«*  =  p2,  then  ( — )  +( - 


104  THEORY  OF  MAXIMA  AND  MINIMA 

x,  x 

Furthermore,  if   xv  ••-,  xn  satisfy   [15],  then,   also,  —  >...,  — 

(x  x  \      1      P  P 

satisfy  these  equations,  while,  since  <£/  -^ ,  •  •  • ,  —  J  =  -^<t>(xv  '  •  *  >  xn}> 

the  signs  of  the  two  quadratic  forms  are  the  same. 

It  is,  therefore,  in  every  respect  admissible  to  add  the  equation 
[16].  We  have,  however,  thereby  gained  an  essential  advantage: 
for  owing  to  the  condition  [16]  none  of  the  variables  can  lie 
without  the  interval  —  1  •  •  •  4- 1 ;  furthermore,  since  the  function 
varies  in  a  continuous  manner,  it  must  necessarily  have  an  upper 
and  a  lower  limit  for  these  values  of  the  variables  xv  x2,  •  • .,  xn; 
that  is,  among  all  systems  of  values  which  satisfy  the  equations 
[15]  and  [16]  there  must  necessarily  be  one*  which  gives  an 
upper  limit  and  one  which  gives  a  lower  limit  of  <£  (see  §  8). 

We  limit  ourselves  to  the  determination  of  the  latter,  By  trial 
we  can  easily  determine  whether  $  reaches  its  lower  limit  on  the 
boundaries,  that  is,  when  one  of  the  x*s  —  ±  1,  while  the  others  are 
all  zero.  If  this  lower  limit  is  not  reached  on  the  boundaries,  then 
(f>  has  a  minimum  value  within  the  boundaries  (cf.  §  64). 

78.  Through  the  addition  of  equation  [16]  the  theorem  of  the 
preceding  article  is  reduced  to  a  problem  in  the  theory  of  maxima 
and  minima;  for  if  the  minimum  value  of  </>(%!>  x%,  •  •  •,  xn)  is 
positive,  <j>  is  certainly  a  definite  positive  form. 

Consequently,  if  we  write 

[17]  G  =  $-  * 

then,  in  order  to  find  the  position  at  which  there  is  a  minimum 
value  of  the  function,  we  have  to  form  the  system  of  equations 

|^=0     (X=l,  2,...,n). 
teA 

O  J  p=Wl          O/J 

This  gives         |E^2i*A+2V<,p~0     (X  =  1,  2, .  .  .,  n), 
™K  rj  fexk 

or, 

[18]  ^AA-  ^A+P1)VV=  0     (X  =  1,  2,  •  •  .,  n). 

M=l  p=l 

*Crelle's  Journal,  Vol.  LXXII,  p.  141 ;  see  also  Serret,  Calc.  diff.  et  int.,  pp.  17  et  seq. 


RELATIVE  MAXIMA  AND  MINIMA 

From  the  n  +  m  +  1  equations 


105 


[19] 


=0     (/>  =  1,  2,  •  •  .,  m), 


the  ?i  +  ??i  H-  1  quantities  a^,  #2,  •  •  •,  xn,fv  £2,  •  •  •,  em,  e  may  be 
determined.  Since  we  know  a  priori  that  a  minimum  value  of 
the  function  <£  in  reality  exists  on  one  position,  we  are  certain 
that  this  system  of  equations  must  determine  at  least  one  real 
system  of  values. 

Consequently  the  first  n  -+-  m  linear  homogeneous  equations  of 
[19]  are  consistent  with  one  another  and  may  be  solved  with 
respect  to  the  unknown  quantities  xv  x2,  •  -  •  ,  xn,  ev  e2,  -  •  •,  em\ 
their  determinant  must  therefore  vanish,  and  we  must  have 


[20] 


^H        £,  ^12>           '  '  'i  AH' 
J                     J              ,,                   A 
xioi  j              "^^22           >             j   "^^2??^ 

«11»  '  '  •>   aml 

a!2»  '  *  •»   ttm2 

alP             «ia»            '  '  •>  aln> 

^••vO 

aml>            am1>           '  *  *'  tt?H?j' 

0,  •  •  -,  0 

=  0. 


The  equation  Ae  =  0  is  clearly  of  the  n  —  mt,h  degree  in  e. 
The  minimum  value  of  (/>  is  necessarily  contained  among  the 
roots  of  this  equation ;  for  if  we  multiply  the  equations  [18] 
respectively  by  xlt  x2,  •  •  •,  xn  and  add  the  results,  we  have 

[21]  <t>(xv  xv  ...,«„)=€, 

it  being  presupposed  that  the  system  of  values  (xv  xz,  •  • .,  xn), 
together  with  the  quantities  ev  €%,••-,  em,  satisfies  the  system  of 
equations  [19],  which  is  only  possible  if  e  is  a  root  of  the  equa 
tion  A  e  =  0.  Furthermore,  among  the  systems  of  values  x  which 
satisfy  the  system  of  equations  [19]  that  system  is  also  to  be 


106  THEORY  OF  MAXIMA  AND  MINIMA 

found  which  calls  for  the  minimum,  and  since  the  value  of  the 
function  which  belongs  to  such  a  system  of  values  is  always  a 
root  of  equation  [20],  it  follows  also  that  the  required  minimal 
value  of  (f>  must  be  contained  among  the  roots  of  this  equation. 
As  already  remarked,  this  minimal  value  must  be  positive  if  $ 
is  to  be  continuously  positive  for  the  systems  of  values  of  the 
x'a  under  consideration,  and  from  this  it  follows  that  Ae  must 
have  only  positive  roots.  For  if  one  root  of  this  equation  was 
negative,  then  for  this  root  we  could  determine  a  system  of 
values  xv  %%,•••,  xn,  ev  e2,  •  •  •,  em  for  which,  as  seen  from  [21], 
<f)  is  likewise  negative. 

Hence,  in  order  that  <j>  be  continuously  positive  for  all  systems 
of  values  of  the  x's  which  satisfy  the  equations  [15],  it  is  neces 
sary  and  sufficient  that  the  equation  A#  =  0  have  only  positive  roots* 

The  question  next  arises,  When  does  the  equation  Ae  —  0 
have  only  positive  roots  ?  It  may  be  answered  in  a  completely 
rigorous  manner  by  means  of  Sturm's  theorem  ;t  but  the  inves 
tigation  is  somewhat  difficult;  and  the  symmetry,  which  we 
especially  wish  to  preserve,  would  be  lost  when  we  applied 
Sturm's  theorem. 

For  develop  the  determinant  according  to  powers  of  e  as 
follows : 

[22]   en-m—B1en-m-1+Bzen-m-2— (-  (-  l)n~m  Bn_m=  0  ; 

then  if  all  the  roots  of  this  equation  are  real  and  positive,  the 
coefficients  B  must  be  all  positive,  and,  reciprocally,  if  the  roots 
of  this  equation  are  real  and  the  B's  are  all  greater  than  0,  the 
roots  of  the  equation  Ae  =  0  are  all  positive.  The  form  is  then 
a  definite  quadratic  form.  The  necessary  and  sufficient  condition 
that  the  form  be  not  a  definite  one  is  that  e  =  0  be  the  smallest 
root  of  the  equation  above. 

*  See  Zajaczkowski,  Annals  of  the  Scientific  Society  of  Cracow,  Vol.  XII  (1867) ;  see 
also  Richelot,  Astronom.  Nachr.,  Vol.  XLVIII,  p.  273. 

t  Burnside  and  Panton,  Theory  of  Equations,  chap,  ix;  Hermite,  Crelle,  Vol.  LII, 
p.  43;  Serret,  Algebre  Sup.,  Vol.  I  (1866),  p.  581;  Kroneeker,  Berlin.  Monatsbericht, 
February,  1873. 


RELATIVE  MAXIMA  AND  MINIMA  107 

79.  We  shall  first  show  that  all  the  roots  of  the  equation 
Ae  =  0  are  real  for  the  case  where  no  equations  of  conditions 
are  present.  (See  J.  Petzval,  Haidinger's  Naturw.  AHh.  II  (1848), 
p.  115.) 

Equation  [20]  reduces  then  to  the  form 


[23] 


-  s> 

=  0,  where  A^u  =  ^ 

A^         ••.,  AHH-e 


an  equation  which  is  called  the  equation  of  secular  variations 
and  plays  an  important  role  in  many  analytical  investigations; 
for  example,  in  the  determination  of  the  secular  variations  of 
the  orbits  of  the  planets,  as  well  as  in  the  determination  of  the 
principal  axes  of  lines  and  surfaces  of  the  second  degree.* 

80.  Weierstrass's  proof  t,  which  is  very  simple,  that  all  the  roots 
of  this  equation  are  real,  depends  only  upon  the  theorem  that  if 
the  determinant  of  a  system  of  n  homogeneous  equations  vanishes, 
it  is  always  possible  to  satisfy  the  equations  through  values  of 
the  unknown  quantities  that  are  not  all  equal  to  zero. 

Instead  of  the  equation  [16]  we  subject  the  variables  to  the 
somewhat  more  general  equation 


where  i/r  denotes  a  homogeneous  function  of  the  second  degree, 
which  is  always  positive  t  and  is  only  equal  to  0  when  the 
variables  themselves  vanish. 

*  In  this  connection  the  reader  is  referred  to  Laplace,  Mem.  de  Paris,  Vol.  II  (1772), 
pp.  293-363;  Euler,  Mem.  de  Berlin  (1749-1750);  Tfieoria  molux  corp.  sol.,  chap,  v 
(1765):  Lagrauge,  Mem.  de  Berlin  (1773),  p.  108;  Poison  et  Hachette.  Journ.  de 
VEcole  Polytechn.,  Cah.  XI  (1802),  p.  170:  Rummer,  Crelle.  Vol.  XXVI,  p.  268: 
Jacob!,  Crelle,  Vol.  XXX,  p.  46:  Christoffel,  Crelle,  Vol.  LXIII,  p.  257  :  Bauer,  Crelle. 
Vol.  LXXI,  p.  40:  Borchardt,  Liouv.  Journ.,  Vol.  XII,  p.  30;  Sylvester,  Phil.  Mag.. 
Vol.  II  (1852),  p.  138;  Salmon,  Modern  Higher  Algebra,  Lesson  VI;  and  see  in  par 
ticular  Edward  Smith,  Solution  of  the  Equation  of  Secular  Variation  by  a  Method  due 
to  Hermite.  (Dissertation.  University  of  Virginia.  1917.)  Numerous  other  references 
are  given  in  the  paper  last  mentioned. 

t  Weierstrass,  Berlin.  Monatsbericht,  May  18,  1868.  Cf.  also  Rrouecker,  Berlin. 
Monatsbericht  (1874),  p.  1. 

t  Note  the  lemma  of  §§  83.  84,  and  85. 


108  THEORY  OF  MAXIMA  AND  MINIMA 

81.  If  we  form  the  system  of  equations  (see  [12]  of  preceding 
chapter) 

[24]  <k-«^A  =  0     (X  =  l,  2,...,W), 

then  these  equations  may  always  be  solved  if  their  determinant 
vanishes. 

This  determinant  is  exactly  the  same  as  that  in  [23]  if  we  write 


We  assume  that  e  =  k  +  li,  where  i  =  V^T,  and  that  we  have  found 
^A  =  ?A  +  ^'     (X  =  l,  2,  •  •  -,n) 

as  a  system  of  values  that  satisfy  the  equations  [24]. 
We  must  consequently  have 


-  (A;  +  to)  ^  (f  j  +  V>  fa  +  v,  •••*,&+  V)  =  0 

(X  =  l,  2,...,  ra). 

Since  the  real  and  the  imaginary  parts  of  these  equations  must  of 
themselves  be  zero,  it  follows,  when  we  observe  that  <£A  and  i/rA 
are  linear  functions  of  the  variables,  that 

<Mfl>  f2»  '  '  •»  fn)~  ^A(fl,  f2»  '  '  'I  fn)+  ^A(^I,  ^72J  '  '  'I  «7n)=  0, 
*A(I?I,  1?2»  •  '  •>  ^n)-  ^A(^I,  ^?2>  •  •  '»  ^)-  ^A(?I,  fa,  •  •  ',  fn)=  0. 

82.  Next  multiply  these  equations  respectively  by  ??A  and  fA, 
take  the  summation  over  them  from  1  to  71,  and  subtracting  one 
of  the  resulting  equations  from  the  other,  then,'  since  (see  [17]  of 
the  preceding  chapter) 


A 

we  have 


or, 


f1,  {„  •  •  -,  ln)}=  0, 
[25]  Jflrfa,  ,„...,  ,„)+  ^r(f      f     .  .  .,  £,)}  =  0. 


RELATIVE  MAXIMA  AND  MINIMA  109 

If  it  is  possible  to  find  systems  of  values  of  the  quantities  xv 
x2,...,  xn  which  satisfy  the  equation  [24]  under  the  assumption 
that  e  =  k  +  li,  then  these  values  must  satisfy  at  the  same  time 
[25]  ;  but  since  after  our  hypothesis  the  quantity  within  the 
brackets  cannot  vanish,  it  follows  that  I  must  be  equal  to  zero; 
that  is,  every  value  of  e  for  which  the  determinant  vanishes, 
is  real. 

Hence  we  have  the  theorem  : 

In  order  that  a  quadratic  form  $(xv  x^,-.-,  xn)  be  invariably 
positive,  it  is  necessar.y  and  sufficient  that  the  development  of  the 
determinant  [23]  ivhich  admits  of  only  real  roots,  when  expanded 
in  powers  of  e,  viz. 

[26]  ev-B^-i  +  Btf*-*  ----  +  (-l)"5n  =  0, 

consist  o/?i  +  l  terms  and  that  these  terms  be  alternately  positive 
and  negative. 

If  the  function  is  to  be  invariably  negative,  then  the  equation 
[26]  must  be  complete  and  have  continuation  of  sign. 

Thus  for  the  case,  where  the  variables  are  subjected  to  no 
conditions  we  have  derived  the  criteria  as  to  whether  or  not  a 
homogeneous  quadratic  form  is  a  definite  one  directly  from  the 
coefficients  of  the  function  and  in  a  form  that  is  perfectly 
symmetric. 

83.  Lemma.  If  a  homogeneous  function  of  the  second  degree 
•^(ajp  x2,  •  •  -,  xn)  can  become  zero  for  any  system  of  real  values 
of  the  variables  which  are  not  all  zero,  then  T/T  may  be  both 
positive  and  negative,  it  being  presupposed  that  the  determinant 
of  ^r  is  different  from  zero. 

Let  the  function  ty  vanish  for  the  system  of  values  (f  v  £2,  •  •  •  ,  f  „) 
and  instead  of  xlf  x2,  •  •  •  ,  xn  write  in  i/r  the  arguments  f  l  +  CjA1, 
f2+  cjc,  •  •  .,  fn+  cnk,  where  the  c's  are  indeterminate  constants. 

Developing  with  respect  to  powers  of  k  we  have 


(f  p  &,•••,£„)+  Aty  (cv  c2  ,  •  •  •  ,  cn).         (i) 


110  THEORY  OF  MAXIMA  AND  MINIMA 

By  hypothesis  the  f  "s  are  not  all  zero,  and  the  determinant  of  i/r 
being  different  from  zero,  it  follows  that  ifra(a=l,  2,  •  •  -,  n)  can 
not  all  be  zero. 

Since,  furthermore,  ca  (a=  1,  2,  •  •  •,  n)  are  arbitrary  constants,  we 

a=  n 

may  so  choose  them  that^c,^^,  f2,  -  -  -,  fw)  is  not  equal  to  zero. 

a  =1 

Now  by  taking  k  sufficiently  small  we  may  cause  the  sign  of 
the  expression  (i)  to  depend  only  upon  the  first  term  on  the 
right-hand  side  of  that  expression. 

Hence,  if  we  choose  k  positive  or  negative,  we  have  systems 
of  values  (xv  x2,  •  •  •,  xn)  which  make  ty  positive  or  negative. 

84.  The  determinant  of  the  system  of  equations  [24]  is  formed 
from  the  partial  derivatives  of 

<f)(xv  x2)  •  .  .,  xn)-e^(x1}  x2,  .  .  .,  xn), 

that  is,  from  <t>a(xv  x2,  .  .  .,  xn)  —  e^a(xv  x2,...,  xn)=  0  (ii) 

(«  =  1,  2,  ...,7i), 

where  <£a  and  tya  denote  -  -^-  and  -  — —  respectively.     If  this 

2  cxa  2>  cxa 

determinant  is  equal  to  zero  for  a  value  of  e,  it  follows  that  we 
can  give  to  the  variables  xv  x2,  -  -  .,  xn  values  that  are  not  all 
zero  and  in  such  a  way  that  the  n  equations  (ii)  exist.  Let  this 
value  of  e  be  e  =  k  +  li]  then  if  I  >  0,  it  may  be  shown  that 
the  function  ^r  can  have  both  positive  and  negative  values. 
Denote  the  system  of  values  (xv  x2,  .  .  .,  xn)  which  satisfy  the 

equation  (ii)  by  fc    ,    .  .,    0  x 

v    /     *     xa=!;a+^r)a      («  =  1,  2, ...,»); 

then,  as  in  §  82,  it  may  be  proved  that 

1  tt  (f v  f j»  •  •  -i  U  +  ^ (iv  ^  •  •  •  i  ^n)]  =  0-  (in) 

Since  by  hypothesis  /  is  not  zero,  the  equation  (Hi)  can  only 
exist  either  when  ^(fj,  |2,  .  .  .,  fn)  and  ^(^  ?72,  .  .  .,  ?;w)  have 
opposite  values  (and  then  it  is  proved,  what  we  wish  to  show, 
that  i/r  can  have  both  positive  and  negative  values),  or  when 
the  two  values  of  the  function  are  both  zero  (and  then  from 
what  was  seen  in  the  preceding  section  T/T  can  take  both  positive 
and  negative  values). 


RELATIVE  MAXIMA  AND  MINIMA  111 

85.  In  this  connection  it  is  interesting  to  prove  the  following 
theorem  :  If  the  determinant  formed  from  the  partial  derivatives 
of  the  homogeneous  quadratic  form  ^(«£p  ^2,  •  •  .,  xn)  is  different 
from  zero,  and  if  among  the  infinite  number  of  quadratic  forms 


X2,  •  •  •,  Xn)+  fJ*^(xv  X2,  •  .  .,  Xn) 

there  is  one  definite  quadratic  form,  the  determinant  formed  from 
the  partial  derivatives  of 

<l>(xv  x2,  .  •  .,  xn)-e^(xv  x2,  .  .  .,  xn) 

vanishes  for  only  real  values  of  e. 

The  theorem  will  also  be  true  if  the  determinant  of  </>  (and 
not  as  assumed  of  T/T)  is  different  from  zero. 

Let  \^  -f  ftji/r  be  a  definite  quadratic  form,  and  write 


We  shall  further  choose  two  constants  X0  and  /*0  in  such  a  way 
that  when  we  put 

<t>(xi>  xv->  xn)> 


c/>  is  different  from  zero. 

We    know    from    the    previous    article    that    the    determinant 
formed   from  the   equations 


can  only  vanish  for  real  values  of  k.     The  equations 

*«-^=0     (a  =  l,2,...,n)  (iv) 

may  be  written  in  the  form 

^Ati)^«=  °      («  =  1,  2,  ..-,  w), 


or  *'=t'^   £°*°    C« =1.2.  •••,*).  («) 

A,Q—  A^A-j 

If  we  eliminate  xv  x2,  •  •  •,  #„  from  these  equations,  we  must 
have  the  same  determinant  for  their  solution  as  from  the  equa 
tions  (iv}. 


112  THEORY  OF  MAXIMA  AND  MINIMA 

Hence  every  k  which  causes  this  last  determinant  to  vanish 
must  also  cause  the  first  determinant  to  vanish.  But  the  &'s  are 
all  real.  It  follows  that  if  we  form  from  them  the  n  expressions 


these  quantities  must  also  be  real. 

Hence  the  determinant  of  the  n  equations 

<l>a-e^a=Q     (a  =  l,  2,.-.  ,w) 
has  always  n  real  roots  e. 

We  may  therefore  say  :  If  among  all  the  quadratic  forms 
which  are  contained  in  the  form 

\(f)(Xv  X2,  •  •  -,  Xn)+  fl^jr(xlt  Xy  '  •  ',  Xn), 

there  is  one  which  can  have  only  positive  or  only  negative  values. 
then  the  determinant  of  </>  —  ety  will  have  only  real  roots,  it  being 
assumed  that  the  determinant  of  (j>  or  of  ty  is  not  zero. 

The  theorem  in  §  80  is  accordingly  proved  in  its  greatest 
generality. 

86.  The  case  where  equations  of  -condition  are  present  may 
be  easily  reduced  to  the  case  already  considered.  The  determi 
nant  [20]  was  the  result  of  eliminating  the  quantities  xv  x2,  •  •  •,  xn, 
«i»  ev  '  •  •>  em  fr°m  tne  n  +  m  equations 

[18]     2  X  A  -  ^  +2<VV*  =  °     (X  «  1,  2,  -.,»), 


[15]  Op  =      ap^=  0     (p  =  1,  2,  . 

ft=i 

Since  the  result  of  the  elimination  is  independent  of  the  way 
in  which  it  has  been  effected,  we  may  first  consider  m  of  the 
quantities  x,  say  :  xv  x2,  •  •  •  xm,  expressed  by  means  of  the  equa 
tions  [15]  in  terms  of  the  remaining  n  —  m  of  the  a?'s,  which 
may  be  denoted  by  %v  £2,  ••-,  fn_m.  We  thus  have 

[27]  ^="~iV^     (M  =  l,2,...,m). 


I 

RELATIVE  MAXIMA  AND  MINIMA  113 

Through  the  substitution  of  these  values,  let  <f>(xv  x2,. . .,  xn) 
be  transformed  into  <l>(£i>  %%>•'•>  Zn-m)  and  the  equation 

2?LA2  =  1  into  VT  (f  1?  f  2, .  .  • ,  f  „  _  m)  =  1. 

The  function  T/T  is  invariably  positive  and  is  only  equal  to  zero 
when  the  variables  themselves  all  vanish. 

The  equations  [18]  may  be  written  in  the  form: 


L  UJLK  p=l      CJC\ 

dx 
Multiplying  these  equations  respectively  by  — -  (X  =  1,  2,  •  • . ,  n), 

and  adding  the  results,  then,  since 


we  have  the  following  equations  : 


The  last  term  of  this  equation  drops  out  if  we  substitute  in  it  the 
expressions  [27],  since  the  0p  expressed  in  the  f's  vanish  identi 
cally,  and  we  have  the  equations 

[28]  f|-^=0     (,  =  l,2,...,^-m). 

V$v  CSv 

Now  give  v  all  values  from  1  to  n  —  m,  and  we  have  a  system  of 
n—  m  linear  homogeneous  equations,  from  which  we  may  eliminate 
the  yet  remaining  £v  £2,  •  •  -,  f,,._TO.  The  result  of  this  elimination 
is  an  equation  in  e  and  must  give  the  same  roots  in  e  as  [20].  The 


% 

114  THEORY  OF  MAXIMA  AND  MINIMA 

equations  [28]  are,  however,  created  in  exactly  the  same  manner 
as  the  equations  [24].  If,  then,  Ae  is  the  determinant  of  these 
equations,  it  follows  that  the  roots  of  the  equation  Ae  =  0  are 
all  real. 

87.  As  the  solution  of  the  theorem  proposed  in  §  77  the  final 
result  is : 

In  order  that  the  homogeneous  function  of  the  second  degree 


be  invariably  positive  for  all  systems  of  values  of  the  quantities 
xv  x2,  •  •  -,  xn,  which  satisfy  the  m  linear  homogeneous  equations 
of  condition  n 


it  is  necessary  and  sufficient  that  the  form  of  the  equation  [20], 
developed  with  respect  to  powers  of  e  and  which  has  only  real 
roots,  consist  of  n  —  m  +  1  terms  and  that  the  signs  associated 
with  these  terms  be  alternately  positive  and  negative.  There  must, 
however,  be  only  a  continuation  of  sign  if  $  is  to  be  invariably 
negative. 

The  above  method  was  first  discovered  by  Lagrange,  who  did 
not,  however,  sufficiently  emphasize  the  reality  of  the  roots  of 
equation  [20]. 

II.     APPLICATION  OF  THE  CRITERIA  JUST  FOUND  TO  THE 
PROBLEM  OF  THIS  CHAPTER 

88.  We  have  determined  the  exact  conditions  necessary  for  a 
homogeneous  quadratic  form  to  be  definite  for  the  case  where  the 
variables  are  to  satisfy  equations  of  condition  and  in  a  manner 
entirely  symmetric  in  the  coefficients  of  the  given  function 
together  with  those  of  the  given  equations  of  condition. 

At  the  same  time  with  the  solution  of  this  problem,  the 
problem  of  maxima  and  minima  which  we  have  proposed  in  this 
chapter  is  solved. 


RELATIVE  MAXIMA  AND  MINIMA  115 

89.  Having  regard  to  the  remarks  made  in  §  71  and  §  74  we 
have  as  a  final  result  of  our  investigations  the  following  theorem  : 

THEOREM.  If  those  positions  are  to  be  found  on  which  a  given 
regular  function  F(xv  x2,  •  •  •,  xn)  has  a  maximum  or  minimum 
value  under  the  condition  that  the  n  variables  xv  x2,  •  •  •,  xn  satisfy 
the  m  equations 

M  /A  fa,  *»  '  •  •>  *„)  =  0      (X  =  1,  2,  •  -  .,  m), 

where  /A  are  likewise  regular  functions,  we  write 


/p/P  =  G  to,  *2, 
p=i 

seek  the  system  of  real  values 


which  satisfy  the  n  -f-  m  equations 

'cG 


If(o>i,  #2>  •  •  -,  an)    is   such   a  system  of  values  of  xl,  x2,  •  •  -,  xn, 
develop  the  difference 

ith  respect  to  powers  of  h,  and  have  (since  no  terms  of  the  first 
dimension  can  appear,  oiuing  to  equations  [c])  the  following 
development : 

=  2^Gn"(al>  CIV  '  '  '»  an)fyA+  '  '  '• 

M!  »' 

We  must  next  see  whether  the  function 

is  invariably  positive  or  invariably  negative  for  all  systems  of 
values  of  the  tis  which  satisfy  the  m  equations 

i    \l      —   n          in  —  1       9.    .        .      vn\ 


[  —  e>   &12>           '  '  '»   ^ln>          /11>  /21>  *  ' 

"»    «/ml 

l»            ^22  ~~  g>  *  '  ">   ^2n>           /12'  /22>  *  ' 

*  '  «/m2 

1,            «»2-           •••.<?„,-«.  /l«-/2»." 

*'  J  mn 

.         /i*          •••>/!,,         °.     0.     '• 

-,  o 

/.•      •••'/„»,      ».   °-   •• 

-,  o 

116  THEORY  OF  MAXIMA  AND  MINIMA 

To  do  this  we  form  the  determinant 


07] 


/ml' 

«TW£  £/m  determinant  put  equal  to  0  is  a%  equation  of  the  m  —  n 
degree  in  e,  which  has  only  real  roots.  Developing  the  determinant 
with  respect  to  powers  of  e,  we  have  to  see  whether  the  develop 
ment  consists  of  n  —  m  + 1  terms  with  alternately  positive  and 
negative  sign  or  with  only  continuation  of  sign. 

If  the  first  is  the  case,  the  function  cf>  is  invariably  positive, 
and  the  function  F  has  on  the  position  (a^,  a2,  •  •  •,  an)  a  minimum 
value ;  if,  on  the  contrary,  the  latter  is  true,  then  </>  is  invariably 
negative,  and  F  has  on  the  position  (av  a2,  •  •  •,  an)  a  maximum 
value. 

This  criterion  fails,  however,  when  <£  vanishes  identically, 
because  the  quantities  <7MV  vanish  for  the  position  (av  a2,  •  .  .,  an); 
and  it  also  fails  when  the  smallest  or  greatest  root  of  Ae  =  0  is 
zero,  since  in  this  case  we  may  always  so  choose  the  h's  that 
<j)  vanishes  without  the  h's  being  all  identically  zero  (see  §  83). 
In  the  latter  case  the  function  </>(#)  is  an  indefinite  or  a  semi- 
definite  form  (§  78). 

In  both  of  these  cases  the  development  [d]  begins  with  terms 
of  the  third  or  higher  dimensions,  and  for  the  same  reason 
as  that  stated  at  the  end  of  §  63  we  cannot  assert  that  in 
general  a  maximum  or  minimum  will  enter  on  the  position 
(alt  a2,  ...,  an). 

90.  We  give  next  two  geometrical  examples  illustrating  the 
above  principles. 

PROBLEM  I.  Determine  the  greatest  and  the  smallest  curvature 
at  a  regular  point  of  a  surface  F(x,  y,  z)  =  0. 


RELATIVE  MAXIMA  AND  MINIMA 


117 


If  at  a  regular  point  P  of  a  plane  curve  we  draw  a  tangent 
and  from  a  neighboring  point  P'  on  the  curve  we  drop  a  perpen 
dicular  P'Q  upon  this  tangent,  then  the  value  that  2  — ^-  =  — * 

pp'2      As 

approaches,  if  we  let  P'  come  indefinitely  near  P,  is  called  the  cur 
vature  of  the  curve  at  the  point  P.    If  the  curve  is  a  circle  with 

radius  r,  the  above  ratio  approaches  —  as  a  limiting  value  and  is, 

r 

therefore,  the  same  for  all  points  of  the  circle.    Now  construct 

the     osculating     circle     which     passes 

through  the  two  neighboring  points  P 

and  P'  of  the  given  curve.    The  arc  of 

the  circle  PP'  may  be  put  equal  to  the 

arc  PP'  of  the  curve,  when  P  and  P' 

are  taken  very  near   each    other,  and 

consequently,  if  r  is  the  radius  of  this 

circle,  the    curvature   of   the   curve  is 

determined  through  the  formula 


[1] 


2  P'Q 
PP'2 


FIG.  11 


The  quantity  r  is  called  the  radius  of  curvature,  and  the  cen 
ter  M  of  the  circle  which  lies  on  the  normal  drawn  to  the  curve 
at  the  point  P  is  known  as  the  center  of  curvature  at  the  point  P. 
The  curvature  is  counted  positive  or  negative  according  as  the 
line  P'Q,  or,  what  amounts  to  the  same  thing,  J/P  has  the  same 
or  opposite  direction  as  that  direction  of  the  normal  which  has 
been  chosen  positive. 

If  we  have  a  given  surface  and  if  the  normal  at  any  regular 
point  of  this  surface  is  drawn,  then  every  plane  drawn  through 
this  normal  will  cut  the  surface  hi  a  curve  winch  has  at  the 
point  P  a  definite  tangent  and  a  definite  curvature  in  the  sense 
given  above. 

The  curvature  of  this  curve  at  the  point  P  is  called  the  curva 
ture  of  the  surface  at  the  point  P=(x,  y,  z)  in  the  direction  of  the 
tangent  which  is  determined  through  the  normal  section  in  question. 


118  THEORY  OF  MAXIMA  AND  MINIMA 

Following  the  definitions  given  above  it  is  easy  to  fix  the 
analytic  conception  of  the  curvature  of  a  surface  and  then  to 
formulate  the  problem  in  an  analytic  manner. 

If  P'  =  (xr,  y',  z')  is  a  neighboring  point  of  P  on  the  surface, 
the  equation  of  the  surface  may  be  written  in  the  form: 


[2]      Q=Fl(x'- 

+  i  Wi 

X'  -  x)  (y'  -  y)  +  2  F^(y'  -  y)  (z'  -  z) 


dF          ,        dF  dF 

where  F\  =  ^>       F^==~^~)       F3=—> 

dx  dy  cz 


-       F     -  F      - 

12      dxdy9      23      dydz        31      dzdx 

The  equation  of  the  tangential  plane  at  the  point  P  is 
[3]  JP1(f-«)  +  ^(i»-y)  +  Ji(r-*)=0, 

where  £,  rj,  £  are  the  running  coordinates. 
Therefore,  if  we  write  for  brevity 


and  take  as  the  positive  direction  of  the  normal  of  the  surface 
at  the  point  P  that  direction  for  which  H  is  positive,  then  the 
direction-cosines  of  this  normal  are 


H' 

Consequently  the  distance  from  P'  to  the  tangential  plane  is 

[5] 


The  negative  or  positive  sign  is  to  be  given  to  the  expression 
on  the  right-hand  side  according  as  the  length  P'Q  has  the  same 


RELATIVE  MAXIMA  AND  MINIMA  119 

or  opposite  direction  as  that  direction  of  the  normal  which  has 
been  chosen  positive. 

In  the   first   case,  paying  attention   to  [2],  which  has  to  be 
satisfied,  since  P1  lies  upon  the  surface,  we  have 

rei 


where  S2  =  (x1  -  xf  +  (yf  -  yf  +  (*'  -  z)2. 

In  the  case  where  the  direction  P'Q  is  contrary  to  the  positive 
direction  of  the  normal,  we  must  give  the  negative  sign  to  the 
right-hand  side  of  [6]. 

Now  let  P1  approach  nearer  and  nearer  P  ;  then  the  quantities 

x'-x       y'-y       z'-z 

-  9  -  j  -  > 

s         s         s 

which  represent  the  direction-cosines  of  the  line  PP't  become  the 
direction-cosines  of  the  tangent  at  the  point  P  of  the  normal  sec 
tion  that  is  determined  through  P'.  Representing  these  by  a,  ft,  7 

P'Q 
and  the  limiting  value  of  2  —  =^-  by  K,  then 

PP' 

[7]     K  =  ~{Fna^F22^  +  F33j^2Fl2aft  +  2F2^j-{-2FB1ya}> 

where  the  terms  of  higher  degree  in  x'  —  x,  etc.  are  neglected.  In 
this  formula  K  represents  the  curvature  of  the  surface  in  the  direc 
tion  determined  by  a,  /3,  7.  This  is  to  be  taken  positive  or  negativ.e 
according  as  the  direction  of  the  length  MP,  where  M  is  the  center 
of  curvature,  corresponds  to  the  positive  direction  or  not. 

If  the  coordinates  of  the  center  of  curvature  are  represented 
by  xQ)  y0,  ZQ  and  the  radius  of  curvature  by  p,  then 

x-x0=p 


H 


or,  since  /c  =  - 

P 


120  THEORY  OF  MAXIMA  AND  MINIMA 


Since  H  does  not  appear  in  these  expressions,  we  see  that  the 
position  of  the  center  of  curvature  is  independent  of  the  choice 
of  the  direction  of  the  normal. 

Suppose  that  the  normal  plane  which  is  determined  through 
the  direction  a,  ft,  7  is  turned  about  the  normal  until  it  returns 
to  its  original  position.  Then,  while  a,  ft,  7  vary  in  a  definite 
manner,  the  function  K  of  a,  ft,  7  assumes  different  values  at 
every  instance,  and  since  it  is  a  regular  function,  it  must  have 
a  maximum  value  for  a  definite  system  of  values  (a,  ft,  7)  and 
likewise  also  a  minimum  value  for  another  definite  system  of 
values  (a,  ft,  7). 

The  quantity  —  has  the  same  value  for  all  normal  sections 
H 

that  are  laid  through  the  same  normal.*  We  have,  therefore, 
to  seek  the  systems  of  values  (a,  ft,  7)  for  which  the  expression 

Fua*  +  F^P+  7^72+  2  Fl2aft  +  2  F23fty  +  2  F31ya 

assumes  its  greatest  and  its  smallest  value. 

We  have  also  to  observe  that  the  variables  a,  ft,  7  must  satisfy 
the  equations  of  condition 


the  first  of  which  says  that  the  direction  which  is  determined 
through  a,  ft,  7  is  to  lie  in  the  tangential  plane  of  the  surface 
at  the  point  P,  while  the  second  equation  is  the  well-known 
relation  among  the  direction-cosines  of  a  straight  line  in  space. 

*  See  Salmon,  A  Treatise  on  the  Analytic  Geometry  of  Three  Dimensions  (Fourth 
Edition),  p.  259. 


KELATIVE  MAXIMA  AND  MINIMA 

Following  the  methods  indicated  in  §  89,  we  write 
[10]      G  =  Flla*+ 


121 


-  1)+  2  e'^a  + 
and  we  then  have  (§89,  [c])  to  form  the  equations 

g=0,  5=0,  g-o, 

da  cp  dy 


from  which  we  must  eliminate  a,  £,  7,  and  «'. 
These  equations  are 


[11] 


where  F^=F^     (X,  ft  =  1,  2,  3). 

Through  elimination  we  have 


=  0, 


[12] 


'13' 


2' 


=  0. 


This  is  an  equation  of  the  second  degree  in  et  and  consequently 
gives  us  two  values  el  and  e2,  which  are  maximum  and  minimum 
values,  since  both  maximum  and  minimum  values  enter,  as 
shown  above.  Multiplying  the  first  three  equations  [11]  by  a,  & 
7  respectively  and  adding  the  results,  we  have 


[13] 


Hence,  from  [7]  we  have 


[14] 


122 


THEOKY  OF  MAXIMA  AND  MINIMA 


Consequently  the  two  principal  curvatures  at  the  point  P  have 
the  values 


[15] 


Pi      H 

I.A; 


and  the  coordinates  of  the  corresponding  centers  of  curvature  are 
found  from  the  formulae 


[16] 


In  order  to  determine  e,  let  us  write 


02 


and  form  from  these  the  corresponding  quantities  through  the 
cyclic  interchange  of  the  indices.  Equation  [12]  may  be  written 
in  the  form* 

Z>ul?  +  D*F*  +  DSZFI  +  2  D12F^  +  2  DuFtFt  +  2  D^F^  =  0. 
Developing  this  expression  with  respect  to  powers  of  e,  we  have 

[17]  H*e*-Le  +  M=Q, 

where    L  =  ff2  (^n  +  F22  +  FBa)  -  (F^F*  +  F^F*  +  FS3Fj) 

+  2  ^i2^2+  2  V2Fg+  2  F^FZF, 
and       M=  (fuFM  -  Fj)  F*  +  (FmF11  -  F,*)  F* 


77T         7yT       \     T7f      77T         i       /  TTf         TJT  T7T          ET       \     T7T      77T 

~  ^31^22)  ^3^1  T  (^31^32  -  ^12^33)  ^1^2' 

From  [17]  we  have  at  once  the  values  of  the  sum  and  the 
product  of  the  two  principal  curvatures,  viz.  (see  equation  [15]): 

1       1        L 


[18] 


PiP*      # 

*  See  Salmon,  loc.  cit.,  p.  257. 


RELATIVE  MAXIMA  AND  MINIMA  123 

We  have  thus  expressed  the  sum  of  the  reciprocal  radii  of  curva 
ture  and  also  the  measure  of  curvature  of  the  surface  at  the  point  P 
directly  through  the  coordinates  of  this  point. 

Although  the  formulae  are  somewhat  complicated,  they  are 
used  extensively  and  with  great  advantage. 

In  the  case  of  minimal  surfaces*  which  are  characterized 
through  the  equation 


we  have  L  =  0. 

This  is  therefore  the  general  differential  equation  for  minimal 
surfaces. 

91  .  PROBLEM  II.  From  a  given  point  (a,  b,  c)  to  a  given  surface 
F(x,  y,  z)  =  0  draw  a  straight  line  whose  length  is  a  maximum 
or  a  minimum. 

Write    G  =  (x-a)*+(y-b)*  +  (z-cf+2\F(x,y)z).  (i) 

Then  the  quantities  x,  y,  z,  \  are  to  be  determined  (see  §  89,  [c]) 
from  the  following  equations: 

x-a  +  \Fl  =  0," 
y-b  +  \F2=Q, 
z  -  c  +\F3=Q, 
F(x,y,z)=0.} 

It  follows,  since  Fv  F2,  F3  are  proportional  to  the  direction- 
cosines  of  the  normal  to  the  surface  at  the  point  (x,  y,  z),  that  the 
points  determined  through  these  equations  are  such  that  lines 
joining  them  to  the  point  (a,  b,  c)  stand  normal  to  the  surface. 

If  P=(x,  y,  z)  is  such  a  point,  then  to  determine  whether  for 
this  point  the  quantity 


is  in  reality  a  maximum  or  a  minimum,  we  substitute  x  H-  u, 
y  +  v,  z  +  w  instead  of  x,  y,  z  in  the  function  G.  The  quantities 
ic,  v,  w  are,  of  course,  taken  very  small. 

*See  papers  by  the  author  on  this  subject  in  the  first  numbers  of  the  Mathematical 
Review. 


124  THEOEY  OF  MAXIMA  AND  MINIMA 

We  must  develop  the  difference 

G(x  +  u,  y  +  v,  z  +  w)-G(x,  y,  z) 

in  powers  of  u,  v,  and  w. 

The  terms  of  the  first  dimension  drop  out,  and  the  aggregate 
of  the  terms  of  the  second  dimension  is 


+  2  F12uv  +  2  F23vw  +  2  F3lwu).  (w) 

Since  the  point  (x  +  u,  y+v,  z+w)  must  also  lie  upon  the  surface, 
the  quantities  u,  v,  w  must  satisfy  the  condition 

F^u,  +  F2v  +  ^3w;  =  0,  (v) 

where  the  terms  of  the  higher  dimensions  are  omitted  (see  [8]  of 
the  present  chapter). 

If  we  wish  to  determine  whether  the  function  -v/r  is  invariably 
positive  or  invariably  negative  for  all  systems  of  values  (u,  v,  w) 
which  satisfy  equation  (v),  we  may  seek  the  minimum  or  maximum 
of  this  function  i/r  under  the  condition  that  the  variables  are  limited, 
besides  the  equation  (v),  to  the  further  restriction  (cf.  [16]  of  §  77) 
that 


For  this  purpose  we  form  the  function 

T/T  -  e(y?+  v*+  w2-  1)+  2  e'^u+Fjo  +F3w), 


and  writing  its  partial  derivatives  with  respect  to  u,  v,  and  w  equal 
to  zero,  we  derive  the  equations 


*  +  (FSS- 


RELATIVE  MAXIMA  AND  MINIMA  125 

Eliminating  u,  v,  w,.—  from  equations  (v)  and  (viii)t  we  have  here 

A, 

exactly  the  same  system  of  equations  as  in  [12]  of  the  preceding 
problem,  except  that  here  -  and  er  stand  in  the  place  of  e  and  e'. 

A 

Denote  the  two  roots  of  the  quadratic  equation  in  e,  which  is 
the  result  of  the  above  elimination,  by  el  and  e%,  and  the  corre 
sponding  radii  of  curvature  of  the  normal  sections  by  p1  and  p2  ; 

then,  since  ^—  -  has  the  same  meaning  as  e  in  the  previous  problem, 


ft 


where  the  positive  direction  of  the  normal  to  the  surface  is  so 
chosen  that  H>Q. 

If  for  the  position  (x,  y,  z)  a  minimum  of  the  distance  is  to 
enter,  then  both  values  of  the  e  must  be  positive  ;  if  a  maximum, 
then  e1  and  e2  must  be  negative. 

It  is  easy  to  give  a  geometric  interpretation  of  this  result  : 
Let  PN  be  the  positive  direction  of  the  normal  and  A  =  (a,  b,  c). 
Then  from  (ii)  it  follows  that  the  length  from  A  to  P  has  the 
same  or  opposite  direction  as  PN,  according  as  X  is  negative 
or  positive. 

Hence,  from  (ii), 

AP=-\H. 

If  the  centers  of  curvature  corresponding  to  p1  and  p.2  be  denoted 
by  ML  and  M2,  then 


,  , 

Hence  e=  and     e 


126  THEORY  OF  MAXIMA  AND  MINIMA 

If,  then,  Ml  and  M2  lie  on  the  same  side  of  P  and  if  A  lies  be 
tween  Ml  and  M2,  as  in  Figs.  12  and  13,  then  the  es  have  different 
signs  and  there  is  neither  a  maxi 
mum  nor  a  minimum.  l *• 

If  Ml  and  M2  lie  on  the  same  side 

of  P  while  A  is  without  the  inter-  P     Mi     A     M 8 

val  M1  •  •  -  M2,  then  a  minimum  or  FIG.  13 

maximum  will  enter  according  as  A 

starting  from  one  of  the  centers  of  1 2        > 

curvature  lies  upon  the  same  side  as 

P  or  not  (see  Figs.  14  and  15).  AM!     M  2    P 

If  the  points  J/j  and  Jf2  lie  on                       FIG.  15 
different  sides  of  P  and  if  A  is  situ 
ated  within  the  interval  Ml  •  •  •  M2,    l »~ 

as  in  Fig.  16,  then  there  is  always 

a  minimum.    If,  however,  A  lies  without  the  interval  Ml  •  •  -  M2, 

then  there  is  neither  a  maximum  nor  a  minimum. 

In  whatever  manner  Ml  and  M2  may  lie,  if  A  coincides  with 
one  of  these  points,  then  one  of  the  two  values  of  e  is  equal  to 
zero,  and  the  general  remark  stated  at  the  end  of  §  89  is  applicable. 

The  above  results  are  derived  in  a  different  manner  by  Goursat, 
Cours  D' Analyse,  Vol.  I,  p.  118. 

The  case  may  also  happen  here  (see  §  72)  that  in  the  solution 
of  the  equations  (ii)  and  (Hi)  a  singular  point  of  the  surface  is 
found  at  the  point  P,  at  which  f\=  0  =  F2  =  F3.  We  cannot  pro 
ceed  as  above,  since,  there  being  no  definite  normal  of  the  surface 
at  such  a  point,  the  determination  whether  for  this  point  a  maxi 
mum  or  minimum  really  exist  cannot  be  decided  in  the  manner 
we  have  just  given. 

The  general  remark  of  §  73  indicates  how  we  are  to  proceed. 

92.  Brand's  problems.  The  two  following  problems  taken  from 
the  theory  of  light  were  prepared  by  my  colleague,  Professor 
Louis  Brand. 

PROBLEM  I.  Reflection  at  the  surface  F(x,  y,  z)  =  0.  A  ray  passes 
from  a  point  J^  to  a  point  P  on  a  given  surface  and  is  reflected  to 
a  point  P^  When  is  P^P  -f-  PP2  &  minimum  ? 


RELATIVE  MAXIMA  AND  MINIMA 
Write  PPl  =  c?!  and  PP^=  d2  so  that 


127 


We  seek  to  find   the  condition  that  makes  f^  +  c?2  an  extreme 
when  P  is  subjected  to  the  condition  of  lying  on  the  surface 

Using  the  Lagrangian  method  (§  89)  we 
must  find  the  extremes  of  the  function 

(x,  y>  *)• 

FIG.  1 


<    x,  y,  *=< 
Writing    £,    for 


etc.,   the 


necessary  conditions  for  an  extreme,  viz.,  <f>x  =  <f>y  =  <f>3  =  0,  give 


[2] 


d. 


Let  the  direction-cosines  of  the  lines  PI[  and  PJ^  be  lv  mv  n^  and 
/2,  m2, 7i2  respectively ;  and  let  /,  m,  n  denote  the  direction-cosines 
of  the  normal  to  the  surface  [1]  at  the  point  P.  Furthermore, 
since  Fxi  FIP  Fz  are  proportional  to  I,  m,  n,  write 

\FX  =  kl,     \Fy  =  km,     \FZ  =  kn. 
Equations  [2]  then  become 


[3] 


m%  =  km, 
nn  =  kn. 


Designate  the  angle  between  PP^  and  PP2  by  (1,  2);  between  PPl 
and  the  normal  by  (1,  n);  between  PP%  and  the  normal  by  (2,  ri). 
It  is  seen  then  that 

cos(l,  2)=  ^/g-f 

cos(l,  n)=  IJ  + 

cos  (2,  n )  =  Z2/  -h 


128  THEORY  OF  MAXIMA  AND  MINIMA 

Multiplying  equations  [3]  by  l^,  mv  nv  respectively,  and  adding, 
it  follows,  since  If  +  m*  +  n*  =  1,  that 

[4]  1  +  cos  (1,  2)  =  k  cos  (1,  ri). 

Similarly,  by  multiplying  equations  [3]  by  /2,  ra2,  w2,  respec 
tively,  and  adding,  we  get 


[5]  cos(l,  2)  +  l  =  &cos(2,  n). 

From  [4]  and  [5]  we  have 

cos(l,  ri)=  cos  (2,  n),  or 
[6]  (1,»)  =  (2,»). 

Moreover,  upon  multiplying  equations  [3]  by  /,  m,  n,  respectively, 
and  adding,  we  get 

cos(l,  n)  -f  cos  (2,  n)=k,  or,  from  [6], 
k=2  cos(l,  n). 

Substituting  this  value  of  ]c  in  [4],  we  have 

l+cos(l,  2)  =2  cos2  (1,  n), 

so  that      cos(l,  2)=  2  cos2(l,  n)-l=  cos  2(1,  n)=  cos  2(2,  n). 
It  follows  that  (1,  2)  =  2  (1,  n)  =  2  (2,  71), 


and  that  the  lines  P.ZJ,  P^,  and  the  normal  must  lie  in  the  same 
plane,  and  it  is  further  seen  that  the  normal  bisects  the  angle 
between  PJJ  and  PP2. 

We  have  thus  arrived  at  the  condition  which  is  an  optical  law  : 
The  incident  and  reflected  rays  must  lie  in  a  normal  plane,  and 
the  angle  of  incidence  must  be  equal  to  the  angle  of  reflection. 

The  above  result  is  merely  a  necessary  condition  for  an  .extreme  ; 
to  find  whether  an  extreme  really  exists,  and  if  it  does,  whether 
it  is  a  maximum  or  a  minimum,  let  us  choose  the  plane  P^PP^  as 
the  #2/-plane. 

If  the  curve  cut  from  the  surface  by  the  plane  I{PI^  has 
the  equation 

[7]  y  =/(*), 


RELATIVE  MAXIMA  AND  MINIMA  129 

the  problem  now  becomes  to  determine  the  nature  of  the  point 

P  which  makes       ,       =  0,  where 
ax 


the  y  being  replaced  by  /(#). 

'  | 


dx  d^  2 

while  the  equation  of  the  normal  to  the  curve  [7]  at  P(x,  y)  is 


and  the  distance  of  the  point  (xiy  y4)  from  this  normal  is 

h  =  (x- 


Further,   take   the    origin    at    the    point  P  and   the    tangent 
to  the   surface    at   P   lying   in   the   plane  Pfl^  as  the  #-axis. 


Then         =  0  shows  that 
dx 


= 


and  as  hl  and  h2  have  opposite  signs,  since  JJ  and  J^  lie  upon 
opposite  sides  of  the  normal, 

sin(l,  TI)=  sin  (2,  n), 

or  (l,7i)  =  (2,  7i), 

as  stated  before  in  [6]. 
Note  that 

dl 
' 


130 


THEORY  OF  MAXIMA  AND  MINIMA 


It  follows  that  for  the  origin  and  the  direction  y'  =  0, 


Writing       0  =  (l,n)  =  (2,n), 

we  note  that  ^1  =  ^  =  cos  0, 
d       d 


^-7^ 


so  that 


=  (-r  +  T-  ]COS20-2?/'cOS0.         FIG.  18 


From  this  it  is  seen  that 


0  ^  0  according  as  y"  =  -(  — ( jcos  6. 

ft /y>&     "^  •*          -^    O  \    /7  /7       / 

« \&\      »a/ 

Since  y'=  0,  we  note  that  y"  is  the  curvature  of  curve  y—f(x)  at  the 

I 
origin,  that  is,  y"  =     >  where  p  is  the  radius  of  curvature.   Hence, 

when 


>  0,  and  the  path  is  a  minimum ;  when 


da? 

[9] 

— Y  <  0,  and  the  path  is  a  maximum. 

To  interpret  this  result  geometrically  it  is  seen  that 


-  — +— icos 
2Wi 

is  the  curvature  of  the  ellipse  whose  foci  are  at  J^  and  P2  and  which 
passes  through  P  (see  Pascal,  Repertorium  der  Hoheren  Mathematik, 
Vol.  II,  1,  p.  245).  The  quantities  d1  and  d2  are  its  focal  radii 
at  P,  and  0  is  the  angle  between  either  focal  radius  and  the  normal 
to  the  ellipse  at  P.  Note  that  this  ellipse  is  tangent  to  the  curve 
[7],  since  the  normal  to  the  curve  bisects  the  angle  between  the 
focal  radii  of  the  ellipse  and  hence  is  also  the  normal  to  the  ellipse. 


RELATIVE  MAXIMA  AND  MINIMA 


131 


When  -  =  -(  —  +  —  )cos0,  the  ellipse  and  the  curve  [7]  have 

P      2\di      **/ 
the  same  curvature  at  P,  and  the  test  for  extremes  is  inconclusive. 

But  here  the  conditions  for  a  maximum  or  a  minimum  are  obvious 
from  geometrical  considerations.  For,  remembering  that  dl  4-  d2 
is  constant  for  points  on  the  ellipse,  say  d1-\-d2  =  k,  then  d1-\-d2<  k 
for  points  within  the  ellipse  and  d1  +  d2  >  k  for  points  without  the 
ellipse.  Hence  the  path  of  the  ray  will  be  a  maximum  or  a  mini 
mum  according  as  the  curve  [7]  lies  within  or  without  the  ellipse 
in  the  neighborhood  of  the  point  P  ;  and  it  is  seen  that  [#]  and  [8] 
are  but  special  cases  of  this  general  condition. 

PROBLEM  II.    Refraction  at  the  surface  F  (x,  y,  z)  =  0.     Using 
the  previous  notation,  it  is  required  to  find  the  conditions  that  make 

the  time  of  passage  from  P^to  P^,  that  is,  —  +  —  >an  extreme,  where 

vl      v2 

v^  and  v2  represent  the  velocity  of  light  in  the  two  media. 
The  Lagrangian  function  is  (§89) 


,  y,z)  = 


\F(x,  yt  z). 


Proceeding  as  in  the  case  of  reflection,  we  find  in  place  of 
equations  [3]  above  Atf 


[1] 


ra, 


=  km. 


=  kn. 


From  these  equations  we  deduce  that 

008(1,  2) 


FIG.  19 


[2] 
[3] 
[4] 


—  + 
»l 


cos(l,  2) 


7 
=  k  cos  (1,  n), 


7 
-f  —  =  k  cos  (2,  n), 


cos(l,  n)      cos (2,  n)      , 

( —  A'. 


132  THEORY  OF  MAXIMA  AND  MINIMA 

Multiplying  [2]  by  —  and  [3]  by  —  >  and  subtracting,  we  have 

Vl  V2 

S(1>  n)      cos (2, 


substituting  from  [4]  the  value  of  k  in  this  equation,  it  is  seen  that 
_!_       1  _cos2(l,  n)      cos2 (2,  n) 

V^        V  V?  V 2 

sin2(l,  n)  _sin2(2,  n) 

It  follows  that 

sin(l,  n)  ^sin(2,  n) 

/?1  /J1 

Vl  V2 

From  [2]  and  [4]  it  is  seen  that 

1      cos(l,  2)      cos2(l,  n)      cos(l,  7i)cos(2,  n) 
1 = 1 > 

Vi  V  V  V 

sin2(l,  n)      cos(l,  7i)cos(2,  n)  —  cos(l,  2) 
or  s = s 

Dividing  this  equation  by  [5]  and  then  multiplying  the  result  by 
sin  (2,  n),  we  find 

sin(l,  7i)sin(2,  n)=  cos(l,  n)  cos  (2,  n)  —  cos(l,  2), 
or  cos  (1,2)=  cos  [(1,  n)  +  (2,  n)]t 

and  therefore 

[6]  (1,2)  =  (l,7i) +  (2, 7i), 

so  that  the  incident  and  the  refracted  ray  lie  in  a  normal  plane. 
Equation  [5]  may  be  put  in  the  form 

sin  (2,  n)  =  v^  =  °' 

where  c  is  the  index  of  refraction  of  the  second  medium  with 
respect  to  the  first  medium.  The  above  is  a  generalization  of  a 
problem  due  to  Fermat. 


RELATIVE  MAXIMA  AND  MINIMA  133 

The  geometrical  criteria  for  a  maximum  or  a  minimum  involves 
a  certain  Cartesian  Oval  whose  foci  are  at  P1  and  P%  and  which 
passes  through  P.  Its  equation  in  bipolar  coordinates  is 


dl  and  d2  being  the  variable  radii  vectores.    For  points  on  this  oval 

—  +  —  is  a  constant,  say  k  ;  for  points  within  this  oval  —  +  —  <  A- 
v\  V2  d  d  v1  v2 

and  for  points  without  this  oval  —  +  -2  >  k. 

v\      V2 
Hence  the  time  occupied  by  the  ray  in  passing  from  P1  to  P  is  a 

maximum  or  a  minimum  according  as  the  curve  cut  from  the  sur 
face  by  the  normal  plane  through  Pl  and  PZ  lies  within  or  without 
this  Cartesian  Oval  in  the  neighborhood  of  the  point  P. 


CHAPTER  VII 

SPECIAL  CASES 

I.    THE  PRACTICAL  APPLICATION  OF  THE  CRITERIA  THAT 

HAVE  BEEN   HITHERTO   GIVEN  AND  A  METHOD  FOUNDED 

UPON     THE     THEORY     OF     FUNCTIONS,     WHICH     OFTEN 

RENDERS  UNNECESSARY  THESE  CRITERIA 

93.  The  practical  application  of  the  established  criteria  is  in 
many  cases  connected  with  very  great,  if  not  insurmountable 
difficulties,  which,  however,  cannot  be  disregarded  in  the  theory. 
For  often  the  solutions  of  the  equations  §  89,  [c],  cannot  be 
effected  without  great  labor,  if  at  all,  and  therefore  also  the  forma 
tion  of  the  function  </>  is  impossible.  It  also  happens,  even  if  the 
function  <f>  can  be  formed,  that  the  discussion  regarding  the  coeffi 
cients  of  Ae  =  0  is  attended  with  much  difficulty.  Moreover,  the 
formation  of  the  function  </>  and  the  investigation  relative  to  the 
coefficients  of  Ae  are  very  often  unnecessary,  since  through  direct 
observation  we  may  in  many  cases  determine  whether  a  maximum 
or  a  minimum  really  exists.  If  it  then  happens  that  the  equations 
[c]  admit  of  only  one  real  solution  (i.e.  of  a  real  system  of  values 
xi>  xi> '  '  '>  xn)>  we  mav  be  sure  that  this  is  in  reality  the  maximum 
or  the  minimum  of  the  function.  In  the  same  way,  if  we  can  con 
vince  ourselves  a  priori  that  both  a  maximum  and  a  minimum 
exist,  and  if  it  happens  that  the  equations  [c]  offer  only  two  real 
systems  of  values,  it  is  evident  that  the  one  system  must  corre 
spond  to  the  maximum  value  of  the  function,  the  other  system 
to  the  minimum  value. 

The  determination  as  to  which  of  the  two  systems  of  values 
gives  the  one  or  the  other  is  in  most  cases  easily  determined. 

One  cannot  be  too  careful  in  the  investigation  whether  on  a 
position  which  has  been  determined  from  the  equations  [a]  and 

134 


SPECIAL  CASES  135 

[c]  of  §  89  there  really  is  a  maximum  or  a  minimum,  since  there 
are  cases  in  which  one  may  convince  himself  of  the  existence  of 
a  maximum  or  a  minimum,  when  in  reality  there  is  no  maximum 
or  minimum. 

For  example,  to  establish  Euclid's  theorem  respecting  parallel 
lines,  one  tries  to  prove  the  theorem  regarding  the  sum  of  the 
angles  of  a  triangle  without  the  help  of  the  theorem  of  the  parallel 
lines.  Legendre  was  able,  indeed,  to  show  that  this  sum  could  not 
be  greater  than  two  right  angles ;  however,  he  did  not  show  that 
they  could  not  be  less  than  two  right  angles.  The  method  of 
reasoning  employed  at  that  time  was  as  follows :  If  in  a  triangle 
the  sum.  of  the  three  angles  cannot  be  greater  than  180°,  then 
there  must  be  a  triangle  for  which  the  maximum  of  the  sum  of 
these  angles  is  really  reached.  Assuming  this  to  be  correct,  it 
may  be  shown  that  in  this  triangle  the  sum  of  the  angles  is  equal 
to  180°,  and  from  this  it  may  be  proved  that  the  same  is  true  of 
all  triangles. 

We  see  at  once  that  a  fallacy  has  been  made.  For  if  we  apply 
the  same  conclusions  to  the  spherical  triangles,  in  the  case  of 
which  the  sum  of  the  angles  cannot  be  smaller  then  180°,  we 
would  find  that  in  every  spherical  triangle  the  sum  of  the  angles 
is  equal  to  180°,  which  is  not  true. 

The  fallacy  consists  in  the  assumption  of  the  existence  of  a 
maximum  or  a  minimum ;  it  is  not  always  necessary  that  an 
upper  or  a  lower  limit  be  reached,  even  if  one  can  come  just  as 
near  to  it  as  is  wished  (see  §  8). 

On  this  account  the  assumption  of  the  existence  of  a  real  maxi 
mum  is  not  allowed  without  further  proof.  We  therefore  endeavor 
to  give  the  existence-proof.  For  this  purpose  we  must  recall 
several  theorems  in  the  theory  of  functions.* 

94.  We  call  the  collectivity  of  all  systems  of  values  which  n 
variable  quantities  xlf  x2,  •  -  -,  xn  can  assume  the  realm  (Gebiet) 
of  these  quantities,  and  each  single  system  of  values  a  position 
in  this  realm.  If  these  quantities  are  variables  without  restric 
tion,  so  that  each  of  them  can  go  from  —  oc  to  +  oc,  we  call  the 

*  Note  especially  §137. 


136  THEORY  OF  MAXIMA  AND  MINIMA 

realm  considered  as  a  whole  (Gesamtgebiet)  an  n-ple  multiplicity 
(n-fache  Mannigfaltigkeit).  If  xv  xz,  •  •  •,  xn  are  independent  of 
one  another,  then  we  say  a  definite  position  (av  a2,  •  •  .,  an)  lies 
on  the  interior  of  the  realm  if  these  positions  and  also  all  their 
neighboring  positions  belong  to  this  region ;  it  lies  upon  the 
boundary  of  the  realm  if  in  each  neighborhood  as  small  as  we 
wish  of  this  position  there  are  present  positions  which  belong 
to  the  realm,  and  also  those  that  do  not  belong  to  it;  it  lies 
finally  without  the  defined  realm  if  in  no  neighborhood  as  small 
as  we  wish  of  this  position  there  are  positions  which  belong  to 
the  defined  region. 

If  the  quantities  xv  #2»  •  •  •,  xn  are  subjected  to  m  equations 
of  condition,  then  we  may  express  these  in  terms  ot'n  —  m  inde 
pendent  variables  ult  u2,  •  •  •,  un_m,  and  the  same  definition  may 
be  applied  to  these  variables. 

95.  The  following  theorems  are  proved  in  the  theory  of  func 
tions:  (1)*  If  a  continuous  variable  quantity  is  defined  in  any 
manner,  this  quantity  has  an  upper  and  a  lower  limit;  that  is, 
there  is  a  definitely  determined  quantity  g  of  such  a  kind  that 
no  value  of  the  variable  can  be  greater  than  g>  although  there 
is  a  value  of  the  variable  which  can  come  as  near  to  g  as  we 
wish.  In  the  same  way  there  is  a  quite  determined  quantity  k  of 
such  a  nature  that  no  value  of  the  variable  is  less  than  Jc,  although 
there  is  a  value  of  the  variable  that  comes  as  near  to  Jc  as  we  wish 
(see  also  §  8). 

(2)t  In  the  region  of  n  variables  xv  x2,  •  •  •,  xnt  suppose  we 
have  an  infinite  number  of  positions  defined  in  any  manner; 
let  these  be  denoted  by  (x[,  x'2,  •  •  -,  x'n).  Furthermore,  suppose  that 
among  the  positions  we  have  such  positions  that  x'n  can  come 
as  near  to  a  fixed  limit  an  as  we  wish.  Then  we  have  in  the 
region  of  the  quantities  xv  x2)  -  •  -,  xn  always  at  least  one  definite 
position  (alt  a%,  •  •  •,  an)  of  such  a  nature  that  among  the  definite 
positions  (x{,  x2,  •  -  •,  x'n)  there  are  always  present  positions  that 

*Dini,  Theorie  der  Functionen,  p.  68.    See  also  a  paper  by  Stolz,  "B.  Bolzano's 
Bedeutung  in  der  Geschichte  der  Infinitesimal  Rechnung,"  Math.  Ann.,  Vol.  XVIII. 
t  Biermann,  Theorie  der  An.  Funk.,  p.  81 ;  Serret,  Calc.  diff.  et  int.,  p.  26. 


SPECIAL  CASES 


137 


The  case  of  a  maximum 


lie  as  near  this  position  as  we  wish ;  so  that,  therefore,  if  8  denotes 
a  quantity  arbitrarily  small, 

x(-a,\<8     (X  =  l,  2,... ,TI). 

This  position  lies  either  within  or  upon  the  boundary  of  the 
denned  region  (x[,  x'2,  •  •  •,  x'n). 

96.  This  presupposed,  let  us  consider  a  continuous  function 
F(xlt  x2,  •  •  •,  xn),  and  let  the  realm  of  the  quantities  xlt  xz,  •  •  •,  xn 
be  a  limited  one,  so  that,  therefore,  we  have  systems  of  values 
which  do  not  belong  to  it.  If  for  every  possible  system  of  values 
(xl9  x2,  -  •  •,  xn)  we  associate  the  corresponding  value  of  the 
function,  which  may  be  denoted  by  xn+1,  then  we  have  denned 
certain  positions  in  the  region  of  n  + 1  quantities.  For  the 
quantity  xn+1  there  is  according  to  the  first 
theorem  an  upper  limit  an+1;  consequently, 
owing  to  the  second  theorem  there  must  be 
within  the  interior  or  upon  the  limits  of  the 
defined  region  a  position  (av  «2,  •  •  •,  ant  a,l+i) 
of  such  a  nature  that  in  the  neighborhood  of 
this  position  there  certainly  exist  positions 
which  belong  to  the  region  in  question. 

Now  if  it  can  be  shown  that  this  position  lies  within  the  interior 
of  the  region,  then  there  is  in  reality  a  maximum  of  the  function 
on  the  position  (av  «2,  •  •  -,  an) ;  on  the  con 
trary,  if  the  position  lies  on  the  boundary, 
we  cannot  come  to  a  conclusion  regarding 
the  existence  of  a  maximum  of  the  func 
tion  x1l+l. 

It  may  in  many  cases  happen  that  one 
can  show,  if  (xv  x2,  •  •  •,  xn)  is  any  position 
on  the  boundary  of  the  realm  and  if  xn+1 
denotes  the  corresponding  value  of  the  function,  that  there  are 
present  within  the  realm  positions  for  which  the  values  of  the 
function  are  greater  than  for  every  position  on  the  boundary.  Then 
the  position  which  we  are  considering  here  cannot  lie  upon  the 
boundary,  and  it  is  clear  that  the  limiting  value  of  the  function 


FIG.  20 


The  case  of  a  minimum 


FIG.  21 


138 


THEORY  OF  MAXIMA  AND  MINIMA 


Case  of  asymptotic  approach 


x     xz 


FIG.  22 


position  Xi 


can  be  assumed  for  a  definite  position  within  the  interior,  since 
the  function  varies  in  a  continuous  manner.  The  analogue  is,  of 
course,  true  for  a  minimum.  If,  however, 
it  does  not  admit  of  proof  that  there  are 
positions  on  the  interior  of  the  defined 
realm  for  which  the  value  of  the  function 
is  greater  or  smaller  than  it  is  for  all 
positions  on  the  boundary,  then  nothing 
can  be  concluded  regarding  the  real  exist 
ence  of  a  maximum  or  a  minimum ;  the 
position  (av  a2,  -  >  •,  an)  would  then  lie 
on  the  boundary  of  the  region,  and  there  might  be  an  asymptotic 
approach  to  the  limiting  value  an  +  1  without  this  value's  being  in 
reality  reached.  Such  cases  need  especial  attention. 

The  figures  give  a  plain  picture  of  what       Maximum  on  the  limiting 
has  been  said  for  the  case  y  =f(x),  where  x 
is  limited  to  the  interval  (x1  •  •  •  x  -  •  •  x2). 

97.  Analogous  considerations  of  the 
above  are  fundamental  in  the  very  defi 
nition  of  an  analytic  function.  For  con 
sider  a  power-series  of  x  assumed  or  given 
in  any  manner ;  let  x1  le  a  definite  value 
of  x.  Then  there  are  three  possibilities: 

(1)  x'  may  lie  in  the  region  of  convergence  of  the  given  series 
or  of  a  series  that  is  derived  (§  138)  from  the  given  series  ;  the  value 
for  x  =  x'  of  this  series  is  a  value  of  the  analytic  function  which 
is  determined  through  the  original  series.     In  other  words,  if  with 
Weierstrass  we  call  the  original  series  as  well  as  any  other  series 
derived  from  this  one  with  regard  to  the  function  which  it  repre 
sents  a  function-element  (Functionenelement),  then   the  first  possi 
bility  consists  in  that,  if  any  function-element  is  given,  the  definite 
value  xf  lies  in  the  region  of  convergence  of  a  function-element 
which   is   derived  from   the  given  one.     We  admit  here  also  the 
complex  variable. 

(2)  It  may  happen  that  x'  does  not  lie  in  the  region  of  con 
vergence  of  any  series  that  has  been  derived  in  this  manner  and 


O 


xz 


FIG.  23 


SPECIAL  CASES  139 

that  we  cannot  derive  from  the  original  function-element  another 
function-element  whose  region  of  convergence  can  come  as  near  to 
the  point  x'  as  we  wish.  In  this  case  the  function  does  not  exist 
for  x  =  x'. 

(3)  Although  we  cannot  find  a  power-series  within  which  x'  lies, 
nevertheless,  it  sometimes  happens  that  we  may  still  derive  elements 
ivhose  regions  of  convergence  contain  positions  which  can  come  as 
near  to  the  point  x1  as  we  wish.  Whether  we  can  then  define  the 
function  for  x  =  x'  by  the  consideration  of  boundary  conditions 
must  in  each  case  be  considered  for  itself. 

If  we  have  case  (1)  before  us,  then  the  function  is  defined 
not  only  for  every  value  x1  but  also  for  all  values  in  the  neigh 
borhood  of  x'  and  has  for  these  values  the  character  of  an  integral 
function. 

The  definition  of  an  analytic  function  as  thus  given  is  prefer 
able  to  other  definitions  from  the  fact  that  the  existence  of 
general  analytic  functions  is  at  once  recognized ;  in  short,  that 
we  have  under  our  control,  in  our  possession,  all  possible  analytic 
functions.  Every  possible  power-series  within  a  region  of  con 
vergence  gives  rise  to  the  existence  of  a  definite  analytic  function. 
Moreover,  one  must  assume  the  duty  of  proving  in  the  case  of 
every  example  that  it  leads  to  just  such  functions. 

For  this  reason  investigations  are  necessary  of  which  formerly 
we  find  no  trace.  If  we  have  a  differential  equation,  we  must 
begin  with  the  proof  that  the  functions  which  satisfy  the  differ 
ential  equation  arise  from  such  function-elements  as  we  have  just 
explained ;  that  is,  we  must  first  show,  if  y  is  the  unknown  func 
tion  and  x  is  the  variable  of  the  differential  equation,  that  this 
equation  can  be  satisfied  through  y  =  P  (x  —  a).  Reciprocally,  if 
any  variable  quantity  y  is  so  connected  with  another  variable 
quantity  x  that  it  satisfies  the  differential  equation,  we  must 
show  that  it  may  be  derived  from  one  single  function-element  in 
the  manner  indicated.  This  last  proof  is  of  especial  importance 
in  the  application  of  analysis  to  geometrical  mechanics. 

When  a  problem  is  given  in  mechanics,  we  have  to  represent 
the  coordinates  of  the  moving  point  as  functions  of  the  time. 


140  THEORY  OF  MAXIMA  AND  MINIMA 

Only  real  values  are  permitted  in  this  problem.  We  cannot 
therefore  a  priori  know  whether  the  required  functions  are 
analytic  or  not. 

These  functions  are  generally  denned  through  differential  equa 
tions.  We  shall  give  the  simplest  case  as  an  example.  Suppose 
we  have  a  system  of  points  that  attract  one  another  according 
to  an  analytic  law,  and  let  xl}  x2,  •  •  •,  xn  be  the  coordinates  of 
these  points.  If  the  motion  is  a  free  one,  we  have  the  differential 
equation  in  the  form 


where  F  denotes  a  given  function  of  xv  #2,  •  •  •  ,  xn.  With  such 
a  problem  we  have  to  prove  before  everything  else  that  the 
required  functions  of  time  are  analytic  functions.  If  for  the 
point  t  =  tQ  the  initial  position  and  the  initial  velocity  are  given, 
then  in  the  neighborhood  of  the  initial  position  we  can  find 
power-series,  and  we  have  to  show  that  through  these  power- 
series  the  required  functions  are  completely  determined. 

II.   EXAMPLES  OF  IMPROPER  EXTREMES  WHERE  THE  DIF 

FERENCE  F(a^  +  hvaz+hz,  .  .  .,  an  +  Jin}  -  F(av  av  .  .  .,  aTO)  IS  NEITHER 

POSITIVE    NOR    NEGATIVE    BUT    ZERO    ON    THE    POSITION 

(av  aa,...,an)  WHICH  IS  TO  BE  INVESTIGATED 

98.  We  shall  now  consider  a  case  which  is  not  included  in  the 
previous  investigations,  but  may  be  in  a  certain  measure  reduced 
to  them:  The  definition  of  the  proper  extremes  of  a  function 
consists  in  the  fact  that  the  difference 

F(al+hl)  «2+  &2,  •  .  -,  an+  hn)  -  F(alt  az,  -  •  •,  an)          (i) 

must  be  invariably  negative  or  invariably  positive.  There  are  cases 
where  an  extreme  does  not  appear  on  the  position  (av  a2,  •  •  •,  an) 
in  the  sense  that  the  above  difference  must  be  positive  or  nega 
tive,  but  in  the  sense  that  the  difference  must  be  zero. 

Suppose,  for  example,  we  have  the  problem:  Determine  a 
polygon  of  n  sides  with  a  given  constant  perimeter  S  whose  area 


SPECIAL  CASES  141 

is  a  maximum,  —  a  problem  which  we  shall  later  discuss  more 
fully  (see  §  101). 

If  this  maximum  is  attained  for  a  definite  polygon,  then  we 
may  at  pleasure  change  the  system  of  coordinates  by  sliding  the 
polygon  in  the  plane  without  altering  the  area. 

For  example,  let  n  =  3,  and  (xv  y^,  (x2,  y2),  and  (xs,  y3)  be  the 
coordinates  of  the  vertices  of  the  triangle.  Then  the  expression 
which  is  to  be  a  maximum  is 


where  the  variables  are  subjected  to  the  condition 

fl  — VW  —  v 

kj  ~    I  tX^n  tX^i 


There  will  not  only  be  one  system  of  values  which  gives  for 
F  a  maximum  value,  but  an  infinite  number  of  such  positions ; 
since,  if  we  take  the  triangle  in  a  definite  position,  we  may  move 
it  in  its  plane  at  pleasure.  This  is  therefore  a  case  where  the 
difference  (i)  is  not  positive  or  negative  but  zero. 

99.  Such  cases,  however,  may  be  reduced  to  maxima  and 
minima  proper  if  we  choose  arbitrarily  some  of  the  variable 
quantities.  In  the  special  example  of  the  preceding  section  we 
may  assume  a  vertex  of  the  triangle  at  pleasure;  let  it  be  the 
origin  of  coordinates,  and  we  further  assume  that  one  of  the 
sides  coincides  with  the  positive  direction  of  the  X-axis,  so 
that  we  may  write  xl=yl=y2  =  0.  If  we  agree  that  the  triangle 
is  to  lie  above  or  below  the  A^-axis,  the  problem  is  completely 
determinate. 

In  so  far  as  the  necessary  conditions  for  the  existence  of  an 
extreme  are  concerned  we  may  proceed  in  precisely  the  same 
manner  as  we  have  hitherto  done,  since  under  the  assumption 
that  there  are  no  equations  of  condition  we  have 

\-hv  a2  +  h2,  •  •  •,  an+hn)-  F(alt  a2,  •  •  -,  an) 
5}  W«(«*  aa,-  •  -,  «„)  +  (&!,  hi,  •  •  -,  kn)&.  (u) 

a=l 


142  THEORY  OF  MAXIMA  AND  MINIMA 

If  a  minimum  is  to  be  present,  then  this  difference  can  never  be 
negative,  but  may  be  zero.  For  this  to  be  possible  the  first  deriv- 

a  =  n 

atives  must  all  vanish.    Since,  if  the  sum  ^haFa(alt  a2,  •  •  >,  an) 

had  (say)  a  positive  value  for  h1=  cv  h2=  c2, .  •  •,  hn=  cn,  then  we 
could  place  ha  equal  to  cji  and  then  choose  h  so  small  that  the 
sign  of  the  right-hand  side  of  (ii)  would  depend  only  upon  the 
sign  of  the  first  term.  If  we  then  make  h  positive  or  negative 
the  difference  would  also  be  positive  or  negative. 

If  equations  of  condition  are  present,  it  may  be  shown,  as 
above,  that  the  derivatives  of  the  first  order  must  vanish,  since, 
if  all  these  derivatives  did  not  vanish,  we  might  express  some 
of  the  tis  through  the  remaining  ones,  and  then  proceed  as  we 
have  just  done.  The  required  systems  of  values  (xv  x2,  •  •  •,  xn) 
will  therefore  be  determined  from  the  same  equations  as  before. 

100.  If  we  have  found  a  system  of  values  of  the  #'s  which 
satisfy  the  equations  of  condition  of  the  problem,  then  in  the 
neighborhood  of  this  position  there  will  be  an  infinite  number 
of  other  positions  which  satisfy  the   equations.    These  last  are 
characterized  by   the  condition   that  the  difference  (i)  vanishes 
identically  for  them. 

This  is  just  the  condition  that  made  impossible  the  former 
criteria,  by  means  of  which  we  could  decide  whether  an  extreme 
really  entered  on  a  position  (alf  «2,  •  •  .,  an)  that  was  determined 
through  the  equations  in  a^,  #2,  •  •  .,  xn. 

One  must  therefore  seek  in  another  manner  to  convince  him 
self  which  case  is  the  one  in  question. 

This  is  further  discussed  in  the  following  problem : 

101.  PROBLEM.   Among  all  polygons  which  have  a  given  number 
of  sides  and  a  given  perimeter,  find  the  one  which  contains  the 
greatest  surface-area.    (Zenodorus.) 

We  see  at  once  that  the  problem  proposed  here  is  of  a  some 
what  different  nature  from  the  problems  of  §§90  and  91,  since 
the  existence  of  the  maximum  value  of  the  function  is  no  longer 
the  question,  as  was  proposed  in  §  49  and  held  as  fixed  through 
out  the  general  discussions.  For  if  the  definition  of  the  maximum 


SPECIAL  CASES  148 

is  such  that  the  function  on  the  position  (alt  az,  •  •  •,  an)  must  have 
a  greater  value  on  this  position  than  on  all  neighboring  positions, 
then  in  this  sense  our  polygon  could  certainly  not  have  a  maximum 
area  ;  since,  if  we  had  such  a  polygon  on  any  position,  we  might 
slide  the  polygon  at  pleasure  without  changing  its  shape  and  con 
sequently  its  area.  Therefore  only  a  maximum  of  the  area  can 
enter,  in  the  sense  that  the  periphery  remaining  the  same  an  in 
crease  in  the  area  of  the  surface  cannot  enter  for  an  indefinitely 
small  sliding  of  the  end-points.  We  consequently  cannot  apply 
our  general  theory  without  further  restriction. 

102.  Let  the  coordinates  of  the  n  end-points  taken  in  a  definite 
order  be  .         ,,..,..  T     y 

1  1>  y\  >  ^2'  y  2  '      >    «>  "«• 

The  double  area  of  a  triangle  which  has  the  origin  as  one  of  its 
vertices  and  the  coordinates  of  the  other  two  vertices  xv  y^  and 
#2,  ?/2  is,  neglecting  the  sign,  determined  through  the  expression 


To  determine  the  sign  of  this  expression  we  suppose  that  the 
fundamental  system  of  coordinates  is  brought  through  turning 
about  its  origin  into  such  a  position  that  the  positive  X-axis  coin 
cides  with  the  length  01.  We  call  that  side  of  the  line  01  posi 
tive  on  which  lies  the  positive  direction  of  the  F-axis  :  The  double 
area  of  the  triangle  012  is  to  be  counted  positive  or  negative 
according  as  it  lies  on  the  positive  or  negative  side  of  the  line  01. 

If  the  point  0  has  the  coordinates  #0,  y0,  the  double  area  of 
the  triangle  is 

2  A012  -  (^  -  XQ)  (2/2  -  y0)  -  (yi  -  T/O)  (,v2  -  #0), 

where  the  above  criterion  with  reference  to  the  sign  is  to  be  applied. 
For  the  polygon  we  shall  take  a  definite  consecutive  arrange 
ment  of  the  points  (1,  2,  •  •  •,  n)  and,  besides,  we  shall  assume  that 
no  two  of  the  sides  cross  each  other.  The  last  hypothesis  is 
justifiable,  since  we  may  easily  convince  ourselves  that  if  two 
sides  cut  each  other  we  may  at  once  construct  a  polygon  whose 
sides  do  not  cut  one  another  and  which,  having  the  same  perim 
eter  as  the  first  polygon,  incloses  a  greater  area. 


144 


THEORY  OF  MAXIMA  AND  MINIMA 


Within  the  polygon  take  a  point  0  =  (x0,  y0)  and  draw  from  it 
in  any  direction  a  straight  line  to  infinity.  This  straight  line 
always  cuts  an  odd  number  of  sides  of  the  polygon. 

Now  if  we  follow  the  periphery  of  the  polygon  in  the  fixed 
direction  (1,  2,  •  •  •,  ?i)  and  mark  the  intersection  of  a  side  by  the 
straight  line  with  -f-  1  or  —  1,  according  as  we  pass  from  the  nega 
tive  to  the  positive  side  of  that  line  or  vice  versa,  then  the  sum  of 
these  marks  is  either  +  1  or  —  1.  In  the  first  case  we  say  that  the 
polygon  has  been  described  in  the  positive 
direction,  in  the  second  case  in  the  nega 
tive  direction. 

It  may  be  proved*  that  whatever  point 
be  taken  as  the  point  0  within  the  poly 
gon  and  in  whatever  direction  the  straight 
line  be  drawn,  we  always  have  the  same 
characteristic  number  +1  or  —  1  if  in  each  case  the  positive 
side  of  the  straight  line  has  been  correctly  determined. 

103.  The  double  area  of  the  polygon  is 


FIG.  24 


2  F=      - 


or 


XQ)  (2/2  -  2/0)  -  fa  -  XQ)  (yl  -  y  0)  +  fa  -  x0)  (y8  -  y  0) 

•  •  +K-«o)  (2/1-2/0)  -(^1-^0 


(a) 


where  the  positive  or  negative  sign  is  to  be  taken  according  as  the 
polygon  has  been  described  in  the  positive  or  negative  direction. 
We  may,  however,  eventually  bring  it  about  through  reverting 
the  order  of  the  sequence  of  the  end-points  that  the  expression 
2  F  is  always  positive. 

104.  Suppose  that  this  has  been  done.  The  function  2F  ia  to 
be  made  a  maximum  under  the  condition  that  the  periphery 
has  a  definite  value  S. 

We  may  write 


where  SA  _lf  A  =     (^A  -  XK  _tf  +  (yA  -  yx  _i)2.  (7) 

*  The  proof  is  found  in  Cremona,  Elementi  di  geometria  projetiva.  Rome,  1873. 


SPECIAL  CASES  145 

Form  the  function 

'  G  =  2F+  e(*1>2  +  s2,3  +  •  •  •  +  sntl-S),  (5) 

and  placing  its  partial  derivatives  equal  to  0,  we  have 


A+1,A  A-1,A 


*A+1,A  *A-1, 

(X  =  1,  2,  •  •  -,  71  ;  however,  for  \  =  n  we  must  write  X  •+•  1  =  1). 

Take  in  addition  the  equation  (/3)  and  we  have  2  n  +  1  equations 
for  the  determination  of  the  2  ?i  -f  1  unknown  quantities 


105.  To    reach    in    the    simplest    manner   the    desired    result 
from  these  equations,  we  adopt  the  following  mode  of  procedure. 

If  we  write  x 

^  =  (<*A-  «^i)+*(jfA-  3h-i),  (?) 

then  zx,  geometrically  interpreted,  represents  the  length  from  the 
point  X  —  1  to  the  point  X  both  in  value  and  in  direction. 
If,  further,  we  write 

z(  =  (a?A  -  XK  ..j)  -  i  (yA  -  yA  _j),  (7;) 

then  *A-4=*A2-i,A-  (^) 

Multiply  the  first  of  equations  (e)  by  t  and  subtract  from  the  result 
the  second  ;  then  owing  to  (?)  we  have 


«A-1.»        •*,»+! 

- 


-^-)=- 

N  SA-1,  A7 


Now,  multiplying  the  last  two  equations  together,  we  have  from  (0) 
and  therefore  s?   ,.  =  «?. 

A  —  1)  A  A)  A  T  1 


146  THEORY  OF  MAXIMA  AND  MINIMA 

106.  Since  sv_lfl,  is  an  essentially  positive  quantity,  it  follows 
that  s  _     =  s  (K] 

and  consequently  the  sides  of  the  polygon  are  all  equal  to  one 

S( 

another.    Hence  each  side  =  -  >  and  we  have  from  U] 

n 

zx  +  -i      ein  4-  S 

— *  =  — =  const. 

If  we  write  2A  =  —  e^1, 

n 

where  <£A  denotes  the  angle  which  sA_1)A  makes  with  the  X-axis, 

then  «<*A  +  i-0A)»=  const, 

or  <£A  +  j  —  c£A  =  const. ;  (X) 

that  is,  all  the  angles  of  the  polygon  are  equal  to  one  another, 
and  consequently  the  polygon  is  a  regular  one. 

It  is  thus  shown  that  the  conditions  which  are  had  from  the 
vanishing  of  the  first  derivatives  can  be  satisfied  only  by  a  regular 
polygon ;  that  is,  if  there  is  a  polygon  which,  with  a  given  perim 
eter  and  a  prescribed  number  of  sides,  has  a  greatest  area,  this 
polygon  is  necessarily  regular. 

Our  deductions,  however,  have  in  no  manner  revealed  that  a 
maximum  really  exists. 

107.  To  establish  the  existence  of  a  maximum  we  must  apply 
the  method  given  in  §§  93-96.    We  note  that  an  upper  limit 
exists  for  the  area  of  the  polygon,  from  the  fact  that  the  number 
of  sides  and  the  perimeter  are  given ;  for  if  we  consider  a  square 
whose  sides  are  greater  than  the  given  perimeter  S,  we  can  lay 
each  polygon  with  the  perimeter  S  in  this  square,  and  in  such 
a  way  that  the  end-points  of  the  polygon  do  not  fall  upon  the 
sides  of  the  square.    Hence  the  area  of  the  polygon  cannot  be 
greater  than  that  of  the  square,  and  consequently  there  must  be 
an  upper  limit  for  this  area,  which  may  be  denoted  by  F0.    The 
question  is,  Can  this  limit  in  reality  be  reached  for  a  definite 
system  of  values?    The  variables  xlt  y^  x2,  y%\  •  •  •;  xn,  yn  being 
limited  to  this  square,  there  must  be  (§  96)  among  the  positions 
(xv  y^  xz,  2/2 ;  •••;  xn>  2/%)  which  fill  out  the  square  a  position 


SPECIAL  CASES  147 

(alf  ^  ;  a2>  &2  ;  •  ••;  an,  bn)  of  such  a  nature  that  in  every 
neighborhood  of  this  position  other  positions  exist  for  which  the 
corresponding  surface  area  F  of  the  polygon  formed  from  them 
comes  as  near  as  we  wish  to  the  upper  limit.  We  may  assume 
that  this  position  is  within  the  square,  since  if  it  lies  by  chance 
on  the  boundary,  then  from  what  has  been  said  above,  it  is 
admissible  to  slide  the  corresponding  polygon  without  altering 
its  shape  and  area  into  the  interior  of  the  square. 

We  assert  that  the  value  of  the  function  F  for  the  position 
(av  &j  ;  «2,  &2;  •  •  •;  an,  bn)  must  necessarily  be  equal  to  F0.  For 
if  this  was  not  the  case,  the  inequality  must  also  remain  if  we 
subject  the  points  av  \\  «2,  62  5  •  •  •  3  an>  bn  to  an  indefinitely 
small  variation  ;  and  on  account  of  the  continuity  of  F  it  would 
not  be  possible  in  the  arbitrary  neighborhood  Qi(av\\  •  •  •  ;  an,  bn) 
to  give  positions  for  which  the  corresponding  area  comes  arbitrarily 
near  the  upper  limit  FQ.  This,  however,  contradicts  the  conclu 
sions  previously  made.  Hence  all  n  corners  with  a  given  periph 
ery  not  only  approach  a  definite  limit  with  respect  to  their 
inclosed  area  but  this  limit  is  in  reality  reached.  Since,  furthermore, 
the  necessary  conditions  for  the  existence  of  a  maximum  have 
given  the  regular  polygon  of  n  sides  as  the  only  solution,  and 
since  we  have  seen  a  maximum  really  exists,  we  may  with  all 
rigor  make  the  conclusion  :  That  polygon  which,  with  a  given 
periphery  and  a  given  number  of  sides,  contains  the  greatest  area 
is  the  regular  polygon. 

PROBLEM 

Among  the  regular  polygons  with  a  constant  periphery,  the  one  with 
the  greatest  number  of  angles  has  the  greatest  area.  (Zenodorus.) 

108.  Hadamard's  problem.  If  Al  =  (xv  yv  z^,  A2  =  (x2,  yz,  22), 
A3  =  (XB,  1/3,  23)  are  the  rectangular  coordinates  of  any  three  points 
from  a  fixed  origin  0,  the  volume  formed  on  the  three  lines  OAV 


>       2/3' 

and  if  x?  +  y?+  z?=  df    (i  =  1,  2,  3), 


148  THEORY  OF  MAXIMA  AND  MINIMA 

where  dv  d2,  d3  are  positive  constants,  it  may  be  easily  shown 
that  A  is  a  maximum  when  A  =  dl  •  d2  -  d3 ;  or,  of  all  parallelo- 
pipedons  constructed  on  the  three  sides  OAV  OA2,  OA3,  the  one 
having  the  greatest  volume  is  the  rectangular  parallelopipedon.  As 
the  parallelopipedon  may  occupy  an  infinite  number  of  positions 
without  changing  the  origin,  we  have  here  a  case  of  improper 
maximum  which  is  of  interest. 

The  extension  of  this  problem  is  due  to  Hadamard.* 


^22> 


where    x%  +  ^|  H h  ^  =  ai  (i=  1,  2,  .  .  -,  w),    £Ae    d's    6em# 

positive  constants,  show  that  the  maximum  of  the  absolute  value 
of  A  is  \  __  fi  J  i 

This  may  be  done  as  follows : 

Let  the  determinant  be  developed  with  respect  to  the  elements 
of  the  iih  line,  so  that 

A  =  AHXH  + Ai2xi2+  •  •  •  +Ainxin.  (i) 

We  then  have  to  find  the  maximum  or  the  minimum  of  the  func 
tion  A  of  the  n  variables  ajn,  xi2,  •  •  •,  xin  which  are  connected 
by  the  relation  ,2  i  2  i  i  ,2  _  ,72  /  •  -\ 

The  Lagrange  method  (§89)  leads  at  once  to  the  conditions 


If  xkl,  xk%,  •  -  •,  xkn  are  the  elements  of  another  line  of  the 
determinant,  we  have 

Anxkl  +  Ai2xk2-\ +Ainxkn  =  0  ;  (iv) 

or,  from  (ii),      xtlxkl  +  xi2xk2-\ +»{»«*»=  0,  (v) 

where  i  =£  k. 

*  Hadamard  (Bull,  des  Sciences  Mathtmatiques,  Second  Series,  Vol.  XVII,  1893). 
Proof  by  Wirtinger  (ibid.,  1908).  An  interesting  application  of  this  problem  is  found 
in  Bocher,  Introduction  to  the  Study  of  Integral  Equations,  pp.  31  et  seq. 


SPECIAL  CASES  149 

From  this  we  conclude  that  the  determinant  can  only  have  an 
extreme  value  when  it  is  orthogonal. 

When  the  conditions  (v)  exist,  the  square  of  the  determinant 
is  another  determinant,  in  which  all  the  elements  are  zero  except 
those  of  the  principal  diagonal,  which  are  df,  d$,  -  •  •  ,  d%. 

It  follows  that  A»  =#.#,...,<*>. 

Here  again  we  have  an  improper  extreme  which  it  is  interesting 
to  consider  further. 

III.  CASES  IN  "WHICH  THE  SUBSIDIARY  CONDITIONS  ARE 
NOT  TO  BE  REGARDED  AS  EQUATIONS  BUT  AS  LIMITATIONS 

109.  Besides  the  problems  already  mentioned,  those  problems 
are  particularly  deserving  of  notice  in  which  the  conditions  for 
the  variables  are  not  given   in   the  form   of   equations   but   as 
restrictions  or  limitations. 

For  example,  let  a  point  in  space  and  a  function  which  depends 
upon  the  coordinates  of  this  point  be  given.  Furthermore,  let  the 
point  be  so  restricted  that  it  always  remains  within  the  interior  of 
an  ellipsoid  ;  then  the  restriction  made  upon  the  point  is  expressed 

through  the  inequality  ^      7,2      -2 

0  =  —  +  —  4-—  <  1 

~  «2  +  52  +  C2 

We  have,  accordingly,  such  limitations  when  a  function  of  the 
variables  is  given  which  cannot  exceed  a  certain  upper  and  a 
certain  lower  limit. 

We  make  such  a  restriction  when  we  assume  that  a  function 
fl  shall  always  lie  between  fixed  limits  a  and  ~b. 

110.  This  limitation,  which  consists  of  two  inequalities 
[a]  *<A<&, 

may  be  easily  reduced  to  one. 

For  from  [a]  it  follows  necessarily  that 

W 

and,  reciprocally,  if  [j3]  exists  and  if  a  <  b,  then  /i  must  be 
situated  between  a  and  b  and  consequently  [a]  must  be  true. 


150  THEORY  OF  MAXIMA  AND  MINIMA 

Every  limitation  of  the  kind  given  may  be  analytically  repre 
sented  as  one  single  inequality  of  the  form  [/3]. 

111.  We  must  next  find  the  algorithm  for  the  cases  under 
consideration.  This  may  be  done  at  once  if  we  consider  that 
such  cases  may  be  reduced  to  those  in  which  occur  equations  of 
condition.  For  this  purpose  we  need  only  establish  the  problem 
of  finding  the  maximal  or  minimal  values  of  a  function  whose 
variables  are  subjected  to  certain  conditions  as  follows: 

It  is  required  among  all  systems  of  values  which  satisfy  the 
equations  /A  =  0(X  —  1,  2,  •  •  .,  m)  to  find  those  for  which  F  is  a 
maximum  or  a  minimum. 

By  proposing  the  problem  in  this  manner,  it  is  clear  that  all 
the  variables  x  which  appear  in  the  equations  of  condition  need 
not  necessarily  be  contained  in  the  function. 

Suppose  further  we  have  the  limitation  that 

[7]  /t>0, 

then,  through  the  introduction  of  a  new  variable  xn+l,  we  may 
transform  this  limitation  into  an  equation  of  condition.  For,  as 
we  have  to  do  with  only  real  values  of  the  variables,  the  equation 

[r]  /*  =  «£« 

denotes  exactly  the  same  thing  as  [7]. 

If,  therefore,  a  function  F(xv  x2,  •  -  .,  xn)  is  to  be  a  maximum 
or  minimum  under  the  limitations 


where  the  /'s  are  functions  of  xv  x2,  >  •  .,  xw  then  we  may  solve 
this  problem  if  instead  of  the  r  last  restrictions  we  introduce  the 
following  limitations  : 

Jm  +  1  =  xn  +  l>  Jm  +  2  ==  Xn  +  2>  *  '  '>  Jm  +  r  ==  %n  +  r- 

The  problem  is  thus  reduced  to  the  one  of  finding  among  the 
systems  of  variables  xv  x2,  •  •  •,  xn  +  r  those  systems  for  which  F 
is  a  maximum  or  a  minimum. 

112.  Examples    of   this   character    occur    very    frequently   in 
mechanics.    As  an  example  consider  a  pendulum  which  consists 


SPECIAL  CASES  151 

of  a  flexible  thread  that  cannot  be  stretched.  The  condition  under 
which  the  motion  takes  place  is  not  that  the  material  point  remains 
at  a  constant  distance  from  the  origin,  but  that  the  distance  can 
not  be  greater  than  the  length  of  the  thread.  Such  problems  are 
more  closely  considered  in  the  sequel.  It  will  be  seen  that  by 
means  of  Gauss's  principle  all  problems  of  mechanics  may  be  reduced 
to  problems  of  maxima  and  minima. 

IV.    GAUSS'S  PRINCIPLE 

113.  For  the  sake  of  what  follows  we  shall  give  a  short  ac 
count  of  this  principle :  Consider  the  motion  of  a  system  of  points 
whose  masses  are  mv  m2,  •  •  •,  mn.  Let  the  motions  of  the  points 
be.  limited  or  restricted  in  any  manner,  and  suppose  that  the  system 
moves  under  the  influence  of  forces  that  act  continuously.  For  a 
definite  time  let  the  positions  of  the  points  and  the  components 
of  velocity  both  in  direction  and  magnitude  be  determined.  The 
manner  in  which  the  motion  takes  place  from  this  period  on  is 
determined  through  Gauss's  principle : 

Let  Av  A.2,  •  •  .,  An  be  the  positions  of  the  points  at  the  moment 
first  considered;  Bv  B^---,  Bn  the  positions  which  the  points 
can  take  after  the  lapse  of  an  indefinitely  small  time  T,  if  the 
motions  of  these  points  are  free;  Cv  <72,  •  •  •,  Cn  the  positions  in 
which  these  points  really  are  after  the  lapse  of  the  same  time  T  ; 
and,  finally,  let  C[t  C&  •  •  -,  C'n  be  the  positions  which  the  points 
may  also  possibly  have  assumed  after  the  time  T,  when  the 
conditions  are  fulfilled. 

If  we  form 

v=n       g  v=n      2 

'„     and     VmJtlCL  , 


it  follows  from  Gauss's  principle  that  from  r  =  0  up  to  a  definite 
value  of  T  the  condition 


[1]  SVAC^SXlW 

v  =  l  ,-  =  1 

v=n       2 

is  always  satisfied ;  that  is?       mvBvCv  must  always  be  a  minimum. 


152  THEORY  OF  MAXIMA  AND  MINIMA 

114.  To  make  rigorous  deductions  from  Gauss's  principle,  which 
was  briefly  sketched  in  the  preceding  section,  we  shall  give  a  more 
analytic  formulation  of  it  :  For  this  purpose  we  denote  the  coordi 
nates  of  Av  by  xv,  yvt  zv)  the  components  of  the  velocity  of  Av  by 
xl>  yl)  zl>  an(i  the  components  of  the  force  acting  upon  Av  by 
Xvt  Yv,  Zv.  The  coordinates  of  Bv  are  therefore 

n  *>O 

*v  +  rxl  +~X»y9+  ryl  +  T-Yv)zv  +  rz'v  +  ~Zv] 
and  from  Taylor's  theorem  the  coordinates  of  Cv  are 

o  29 

xv  +  rxl  +  ^xl'  +  .  .  .,  y  T 

consequently  we  have 
[2] 


Instead  of  x",  however  (see  preceding  section),  other  values  may 
possibly  enter,  say  x"+  £„,-••,  so  that  we  have 


rqn       X^,™    »  n'    _  W    //^''_i_  t         "F\2 
L^J       ^mvA,%    s  JMMW  T  &  "- -AT) 

v  =  l  v  =  l 

It  follows  from  Gauss's  principle  that  the  difference  of  the  sums 
[2]  and  [3]  must  always  be  positive. 
Hence 

[4]     0  >jj?m.  \  2  [fc,(av"  -  X.)  +  rjv  (yl1  -  Yv)  +  £  (*„"  -  Zv)] 

+  ?v2  +  ^2-l-C2?j  +  ...; 

that  is,  the  quantities  x" t  yl' ,  z!J  must  be  such  that  the  sum  [2] 
is  a  minimum. 

Hence,  among  all  the  x",  y" ,  z"  which  are  associated  with 
the  conditions  of  motion,  we  must  seek  those  which  make  [2] 
a  minimum. 

115.  We  have  reached  our  proposed  object  if  we  can  show 
that  the  ordinary  equations  of  mechanics  may  be  derived  from 
Gauss's  principle. 


SPECIAL  CASES  153 

If  there  are  no  equations  of  condition  present,  then  clearly  [2] 
is  only  a  minimum  when 

r ''  —  Y    11 "  —  V    z"  —  Z 
<LV  —  -A-v)  yv  —  -*-!/)  2f  —  ^v 

If,  however,  we  have  equations  of  condition,  for  example,  if  any 
of  the  variables  are  connected  by  a  relation  such  as  /  (x,  y,  z)  =  0, 
then  these  must  hold  true  throughout  the  whole  motion.  They 
may  therefore  be  differentiated.  We  have  in  this  way  equations 

in  _^,  _J^,  anci  _?*.    Differentiate  again  and  we  have  equations 
dt      dt  dt 

in  a;,",  y'v' t  and  z" ; 

Hence,  in  conformity  with  the  rules  that  have  been  hitherto 
found  for  the  theory  of  maxima  and  minima,  the  quantities 
xl'>  Uv>  zv  are  t°  be  so  determined  that  the  derived  equations 
of  condition  are  satisfied,  while  at  the  same  time  [2]  becomes  a 
minimum.  But  in  this  case  also,  as  is  easily  shown,  we  are  led 
to  the  usual  differential  equations  of  mechanics. 

116.  The  theory  of  maxima  and  minima  may  be  applied 
to  realms  which  are  seemingly  distant  from  it.  An  example 
in  question  is  the  proof  of  a  very  important  theorem  in  the 
theory  of  functions. 

THE  EEVERSION  OF  SERIES 
If  the  following   n   equations    exist    among    the    variables  xlt 


-  •  •  •       a1  nxn 
=  a21x1 


yn=anlxl+an 

where  the  coefficients  on  the  right-hand  side  are  given  finite  quan 
tities  and  the  "JT's  are  power-series  in  the  x's  of  such  a  nature 
that  each  single  term  is  higher  than  the  first  dimension,  and 
if  the  series  on  the  right-hand  side  are  convergent  and  the 


154 


THEOKY  OF  MAXIMA  AND  MINIMA 


determinant   of  the    nth    order    of   the    linear  functions    of   the 
x's  which  appear  in  [1],  namely, 


[2] 


>  aln 


n2> 


is  different  from  zero,  then,  reciprocally,  the  x's  may  also  be 
expressed  through  convergent  series  of  the  n  quantities  y  which 
identically  satisfy  the  equations  [1]. 

117.  As  an  algorithm  for  the  representation  of  the  series  for 
the  x's,  we  make  use  of  the  following  methods  (cf.  §§  135,  136) : 

If  we  solve  the  equations  [1]  linearly  by  bringing  the  terms  of 
the  higher  powers  of  the  x's  on  the  left-hand  side,  we  have 


where  A^  denotes  the  corresponding  first-minor  of 
It  is  seen  that  in  general 

>L~n  <x  =  M  A 


[3]  x.  = 


M  in  [2]. 

r  1(2) 
A 


where  [xv  x%,  •  •  •,  xn]@)  denotes  terms  of  the  second  and  higher 
dimensions  in  xv  x%,  •  •  •,  xn. 

We  shall  therefore  have  a  first  approximation  to  the  result  if 
we  consider  only  the  terms  on  the  right-hand  side  of  [3]  which 
are  of  the  first  dimension.  A  second  approximation  is  reached 
if  we  substitute  in  the  right-hand  side  of  [3]  the  first  approxima 
tions  already  found  and  reduce  everything  to  terms  of  the  second 
dimension  inclusive.  Continuing  with  the  second  approximations 
that  have  been  found,  substitute  them  in  [3]  and,  neglecting  all 
terms  above  the  third  dimensions,  we  have  the  third  approxi 
mation,  etc.  ;  we  may  thus  derive  the  x's  to  any  degree  of 
exactness  required. 


SPECIAL  CASES  155 

Since  A  is  found  in  all  the  denominators,  the  development 
converges  the  more  rapidly  the  greater  A  is. 

118.  In  what  follows  we  shall  assume  that  the  quantities  on 
the  right-hand  side  of  [1]  are  all  real  and  that  we  may  write 


where  H^  is  a  homogeneous  function  of  the  ith  degree  in  xv 
X2> '  '  '>  xn>  an(^  consequently  by  Euler's  theorem  for  homogeneous 
functions 

-_-._.  GJj[\n  Cjtl\n  I  VJrL\f) 

^L'  rp  A  &       I  ™  *&       I  I       ,y,  A  ^ 

^x  ~  9     1  ~?7"  ^02  -oTT"  ^  O  ^»    ^r 

^  t/'X/j  Zi  c/t/^2  *  l/i*/«. 

+  o  ^1  ~^T"  +  o  ^2  -oT~  "•"  h  o 

O  Vtb-t  tj  (siA/n  <-) 

+        ™  ^4    i         ™  ^4    i  i „, 

T  *l  "^  r  T  ^2  ~^          T"  •  •  •  T  T*n 


J.    J    ^7-  4.    J    fir  4-         ^r 

•i  CXt  4  £<£„  ^±  6/Xr 


where  the  quantities  X  (X  =  1,  2,  •  •  •,  n  ;  /u  =  1,  2,  •  •  •,  n)  are 
continuous  functions  of  the  sc's,  which  become  indefinitely  small 
with  the  aj's. 

The  system  of  equations  [1]  may  then  be  brought  to  the  form 


M  =1 

The  theorem  of  §  116  in  this  modified  form  may  be  expressed 
as  follows  : 

(1)  It  is  always  possible  so  to  fix  for  the  variable^  xlt  &2,  •  •  •,  xn 
and  yv  y2,  -  •  •,  yn,  limits  gv  g2,  —  -,gn  and  hv  &2,  •  •  •,  hn  that  for 
every  system  of  the  y's  for  which  y\\<h^  there  exists  *  one 
system  of  the  x's  for  which  XK  \  <  gK,  and  in  such  a  way  that 
the  equations  [1#]  are  satisfied. 

*  See  Biermann,  Theo.  der  An.,  Funk.,  p.  234,  and  also  Stolz,  p.  172. 


156  THEORY  OF  MAXIMA  AND  MINIMA 

(2)   The  solution  of  the  equations  [la]  has  a  form  similar  to 
the  equations  [la]  themselves,  viz.: 


where  the  Y     are  continuous  functions  of  the  y's,  which  become 
indefinitely  small  with  these  quantities. 

To  prove  this  theorem  we  make  use  of  the  theory  of  maxima 
and  minima. 

119.  If  we  give  to  the  yK  the  value  zero,  the  equations  [la] 
can  only  be  satisfied  if  their  determinant  vanishes,  that  is,  when 


except  for  the  case  where  the  a?'s  vanish. 

For  sufficiently  small  values  of  the  x's  the  determinant  [4]  is 
not  very  different  from  the  determinant  [2].  We  may  therefore 
determine  limits  g  for  the  x's  so  that  [4]  cannot  be  zero  unless 
[2]  is  also  zero.  A  —  0  is,  however,  by  hypothesis  excluded. 
Accordingly  the  y's  can  only  be  zero  in  [la]  when  all  the  x's 
vanish,  provided  the  x's  are  confined  within  fixed  limits.  These 
limits  may  be  regarded  as  the  boundaries  of  a  definite  realm. 

120.  Again,  we  write 


=2Xax*  +XJ  *>  = 
(X=l,  2,.. 


and  consider  the  function 
[6]  5 


A=l 

In  $  we  shall  write  for  the  x's  all  the  systems  of  values  where 
at  least  one  x  lies  on  the  boundary  of  the  realm  in  question. 
The  realm  of  the  x's  is  thus  the  totality  of  the  x's  for  which 


A      S  Zer° 


(X  =  l,  2,  -  •-,%;  /*=!,  2,...,w); 

it  follows  then  that  [4]  is  not  zero,  since    aAJ  is  by  hypothesis 
different  from  zero. 


SPECIAL  CASES  157 

When  one  of  the  x's  reaches  its  limit,  there  is  no  system  of 
values  of  the  x's  for  which  the  function  [6]  vanishes,  since  the 
function  can  (as  follows  from  definition  [5]  of  the  F^  and  the  con 
siderations  of  §  119)  only  vanish  if  all  the  y's  and  consequently 
all  the  x's  vanish. 

There  is  then  a  lower  limit  G  which  is  different  from  zero 
for  the  values  of  [6]  which  correspond  to  a  definite  system  of 
values  (xv  x%,  •  •  •,  xn)  of  the  boundaries. 

121.  We  come  next  to  the  determination  of  the  limiting  values 

of  the  y's.    For  this  purpose  we  consider  the  expression 

» 

m 

If  we  ascribe  definite  values  to  the  y's,  then  there  is  for  the  values 
[7]  in  the  realm  of  the  x's  a  system  for  which  [7]  is  a  minimum. 

We  wish  to  show  that  this  system  of  values  of  the  x's  does 
not  lie  upon  the  boundary  of  the  realm.  We  prove  this  by  show 
ing  that  there  is  a  point  within  the  realm  where  the  expression 
[7]  has  a  smaller  value  than  it  has  on  the  boundary. 

The  expression  [7]  may  be  written 


Since     $  is  at  all  events  greater  than  Fk,  and  consequently 

A<>. 

it  follows  that  vs 

<  ^  ix  <  ^  fix  i  <  ^  SI  y. 


where  the  h's  are  the  limits  of  the  ^'s.    From  this  it  results  that 


-n         ju.  =  n 


and,  consequently,  for  a  greater  reason 
[8] 


H  =  l 


158  THEORY  OF  MAXIMA  AND  MINIMA 

The  limits  h^  must  be  so  chosen  that  the  right-hand  side  of 
[8]  is  positive.  This  choice  can  be  made  so  that  the  expression 
on  the  right-hand  side  for  a  system  of  x's  which  belongs  to  the 
boundary  does  not  become  arbitrarily  small  but  always  remains 
greater  than  a  certain  lower  limit  (see  the  preceding  section). 

The  expression,  however,  on  the  interior  of  the  realm  of  the 
x's  may  be  arbitrarily  small,  viz.,  when  x1  =  x%  =  •  •  •  =  xu  —  0. 

For  this  system  of  values  the  left-hand  side  of  [8]  is  equal  to 


We  have  therefore  found  the  following  result:  We  can  give 
limits  g  to  the  variables  x,  and  to  the  y's  the  limits  h,  in  such 
a  way  that  the  expression  [7]  for  systems  of  values  of  the  x's 
which  belong  to  the  boundary  of  the  realm  is  always  greater  than 
it  is  for  the  zero  position  (xl  =  x2  —  •  -  •  =  xn  =  0). 

Hence  the  position  for  which  the  expression  [7]  is  a  minimum 
must  necessarily  lie  within  the  realm  of  the  x's ;  and  we  may  be 
certain  that  within  the  realm  of  the  x's  there  is  a  position  where 
[7]  has  its  smallest  value. 

122.  In  order  to  find  the  minimal  position  of  [7]  which  was 
shown  to  exist  in  the  previous  section  we  must  differentiate  the 
function  [7]  and  put  the  first  partial  derivatives  equal  to  0. 

This  gives 

•>  =  !  X* 

These  n  equations  can,  in  case  the  determinant 


[10] 


(»  =  l,2,.-.,n;  /t=l,  2, ...,») 


is  different  from  zero,  exist  only  if  the  quantities  within  the 
brackets  vanish. 

[10]  is  identical  with  the  determinant  [4];  and  (see  §  119) 
it  may  be  always  brought  about  through  suitable  choice  of  the 
limits  g  of  the  x's  that  [4]  is  different  from  zero  if  only  the  de 
terminant  [2],  as  by  hypothesis  is  the  case,  is  different  from  zero. 


SPECIAL  CASES  159 

Hence  the  equation  [9]  can  only  be  satisfied  if 

[la]  or  [5]  yv=Fv(xv  x^  .  .  .,  xn). 

We  have  therefore  found  that,  since  there  is  certainly  a  system 

v=n 

of  values  of  tJie  x's  for  which  the  function  ^(yv  —  Fv)z  is  a  mini- 

v  =  l 

mum,  there  must  also  be  a  system  within  the  realm  of  the  x's  for 
•which  the  equations  [1#]  are  satisfied  if  to  the  y's  definite  values 
in  their  realm  are  arbitrarily  given. 

123.  We  must  further  see  whether  within  the  fixed  realm 
there  is  one  or  several  systems  of  values  of  the  x's  that  satisfy 
the  equations  [1#]  with  prescribed  values  of  the  y's  which  lie 
within  definite  limits. 

To  establish  this,  assume  that  (x[,  x%,  •  •  •,  x'n)  is  a  second 
system  of  values  that  satisfy  the  equations  [!«];  we  must  then 
have  the  equations 

[11]  Fv(x[,xi,  •  •  -,  x'n)-Fv(xv  x2,  .  .  .,  ^)  =  0      (v  =  l,  2,  .  .  .,  n). 

Developing  by  Taylor's  theorem,  we  have,  when  we  consider  only 
terms  of  the  first  dimension, 


The  X  are  functions  which  depend  upon  the  x"s  and  x's  and 
vanish  with  these  quantities. 

We  may  determine  the  n  unknown  quantities  x[  —  xlt  x^  —  x2, 
•  -  .,  x'n—  xn  from  the  n  linear  equations  [11  a]. 

For  small  values  of  x  and  xr  the  determinant 


will  be  little  different  from  the  determinant  [10]. 

We  may  therefore  make  the  limits  g  of  the  x's  so  small  that  [12] 
is  different  from  zero  for  all  the  x*a  and  x"s  which  belong  to  the 
realm  ;  and  when  this  has  been  done,  the  equations  [11  a]  are  only 

satisfied  for  r'_r*   //,—  1    9  «v 

xp  —  Zp     (p  —  -L,  A  •  •  •  ,  n)  , 

that  is,  there  exists  within  the  realm  in  question  no  second  system 
of  the  x's  which  satisfies  the  equations  [!«]. 


160  THEORY  OF  MAXIMA  AND  MINIMA 

We  have  therefore  come  to  the  following  result  : 
It  is  possible  so  to  determine  the  limits  g  and  h  that  with  every 
arbitrary  system  of  the  y's  in  which  each  single  variable  does  not 
exceed  its  definite  limiting  value,  the  given  equations  [1«]  are 
satisfied  by  one  and  only  one  system  of  the  x's  in  ivhich  these 
quantities  likewise  do  not  exceed  their  limits* 

The  first  part  of  the  theorem  given  in  §  118  is  thus  proved. 

REMARK.  We  have  assumed  that  we  have  to  do  only  with  real  quantities. 
The  discussion,  however,  is  not  restricted  to  such  quantities,  as  it  is  easy 
to  prove  that  the  same  developments  may  also  be  made  for  complex 
variables. 

124.  The  values  of  the  x's  which  were  had  from  the  equations 


may  be  derived  in  the  manner  given  in  §  118. 
If  we  write 

|^P+X,P|=  A     ("=  1,  2,  •  •  -,  n  ;  p  =  1,  2,  .  .  .,  n), 
the  linear  solution  of  the  equations  [la]  is 

[36]  •V*£'^f9,     (p  =  l,2,...,»), 

„=!  A 

where  A'vp  denotes  the  corresponding  first  minors.  Now  A'  is  a 
definite  quantity  which  lies  within  certain  finite  limits  ;  the  same 

is  also  true  of  —  •    Arvp  is  found  in  a  similar  manner.     Hence 

A 
J 
the  quantities  —  ^  are  finite  quantities  which  lie  between  definite 

limits  ;  and,  therefore,  if  the  y's,  become  indefinitely  small,  the 
x's  will  also  become  indefinitely  small;  that  is,  those  systems  of 
values  of  the  x's  which  satisfy  the  equations  [1]  under  the  named 
conditions  are,  as  has  also  been  shown  in  §  119,  so  formed  that 
they  become  indefinitely  small  with  the  y's. 

*  See  also  Hadamard,  "  Sur  les  transformations  ponctuelles,"  Bull,  de  la  SocitU 
Math.,  Vol.  XXXIV,  1906. 


SPECIAL  CASES  161 

We  may  now  show  that  the  x's  are  continuous  functions  of 
the  ?/'s. 

Let  (ftp  62,  •  •  •  ,  £>„)  be  a  definite  system  of  values  of  the  y's, 
and  let  the  system  (av  «2,  •  •  •,  an)  of  the  x's  correspond  to  this 
system  of  the  i/'s. 

If  we  then  write 

noi  \y\=  ^x  +  ^xl       ~       .,    0 

[13]  I      (X=l,  2,...  ,71), 

l«A=«A+fAj 

the  system  of  equations  [la]  or  [1]  goes  into 


(X  =  l,  2,...  ,TI]. 
Developing  this  expression  according  to  powers  of  the  f  's,  we  have 


where  the  C1^  are  functions  of  the  #'s  and  f's.  If  the  f's  are 
indefinitely  small,  we  may  limit  the  C1^  to  the  first  derivatives 
of  F^.  In  this  case  we  denote  the  coefficients  of  [Ic]  by  (7AM,  so  that 

rT? 

<?AM  =  T^  for  (^1  =  al»  ^2  =  a2>  '  '  •>  ^w  =  a«) 
0gp 

(X  =  l,  2,  •  •  .,  ?i;  /*  =  !,  2,  •  •  -,  ?i), 
and  the  determinant  of  the  equations  [Ic]  goes  into 

for  (xl  =  av  x2  =  aa,  .  .  .,  xn  =  an) 

(\  =  1,  2,...,  7^;  ^=l,2,..-,n). 

If  the  #'s  lie  within  definite  limits,  this  determinant  remains 
always  above  a  definite  limit.  We  may  therefore  say  that  the 
determinant  has  a  value  different  from  zero.  Consequently  the 
condition  that  the  equations  [Ic]  may  be  solved  is  satisfied,  and 
it  is  seen  that  indefinitely  small  values  of  the  f  's  must  correspond 
to  indefinitely  small  values  of  the  TJ'S. 

This  means  nothing  more  than  that  the  functions  x  are  con 
tinuous  functions  of  the  *s. 


162  THEORY  OF  MAXIMA  AND  MINIMA 

125.  The  above  investigations  are  true  under  the  assumption 
that  the  functions  F^  are  continuous,  that  their  first  derivatives 
exist  and  likewise  are  continuous  within  certain  limits.  We  need 
know  absolutely  nothing  about  the  second  derivatives. 

Of  the  #'s,  of  which  it  is  already  known  that  they  exist  as 
functions  of  the  y's  and  vary  in  a  continuous  manner  with  them, 
it  may  now  likewise  be  proved  that  they,  considered  as  functions 
of  the  y's,  have  derivatives  which  are  continuous  functions  of 
the  7/'s. 

The  proof  in  question  may  be  derived  from  the  following  con 
siderations  :  If  from  [Ic]  we  express  the  fs  in  terms  of  the  ?;'s, 
we  have 


The  —  ^  are  continuous  functions  of  the  f's,  and  the  f  s  are 

continuous  functions  of  the  ?;'s.    Hence  —  ^  may  be  represented 
as  continuous  functions  of  the  rfs. 

If   the    rj's   become   indefinitely    small,    then    the   f's   become 

indefinitely  small,  and  we  have  definite  limits  for  —^. 

In  general,  if  we  have  a  function  f(xv  •  •  •  ,  xn)  of  the  n  variables 
#i>  %%>  '  •  •>  xn>  and  if  we  consider  the  difference 

/(«i  +  fi>  «2  +  fa»  '  '  '»  a»  +  f»)  -f(av  av"  '>  an)> 
it  is  seen  that  it  may  be  written  in  the  form 


where  the  ^TA  depend  upon  the  f's  and  become  indefinitely  small 
with  these  quantities,  and  the  &A  are,  in  virtue  of  the  definition 
of  the  differential  quotient,  the  partial  differential  quotients  of  / 
with  respect  to  #A  for  the  system  of  values  (av  a2,  •  •  •,  an).  From 
the  above  it  results  not  only  that  the  x*s  are  continuous  func 
tions  of  the  y's  but  also  that  the  derivatives  of  the  first  order 
of  these  functions  exist. 


SPECIAL  CASES  163 

We  have,  indeed,  the  derivatives  of  the  first  order  if  in  the 
expressions  —  ^  we  write  the  f's  equal  to  zero. 

The  quantities  —  ^,  however,  become  then,  in  accordance  with 

[Ic],  the  quantities  which  we  should  have  in  [Ic]  if  we  had  at 
first  written  C^  instead  of  CAM. 

But  the  quantities  C^  are  continuous  functions  of  alt  a2,  •  -  .,  an. 

We  may  therefore  say  that  the  differential  quotients  —  ^  are  con- 

A 

tinuous  functions  of  the  variables  xt  and  it  is  then  proved  that 
the  x's  are  such  functions  of  the  y's  as  the  y's  are  of  the  x's. 

126.  For  the  complete  solution  of  the  second  part  of  the 
theorem  in  §  116  we  have  yet  to  show  that  the  expressions  [36] 
may  be  reduced  to  the  form  [3  a].  , 

For  this  purpose  we  must  bring  the  quantities  —  ^f  in  [36] 
(§  124)  to  the  form  , 

~ 


where  6Aja  is  the  value  of  -**&  when  all  the  x's  are  equal  to  zero. 

YA/X  is  a  function  of  the  x's,  but  the  x's  are  functions  of  the  y's,  so 
that  YA   is  a  function  of  the  y's  which  vanishes  when  they  vanish. 
We  may  therefore  in  reality  write  [36]  in  the  form  [3  a] 


A  —  1 

127.  There  may  arise  cases  hi  which  we  know  nothing  further 
of  the  functions  F^,  as  was  assumed  in  §  116,  than  that  they  are 
real  continuous  functions. 

We  cannot  then  conclude,  for  example,  that  the  x's  may  be 
developed  in  powers  of  the  y's  ;  but  we  may  reduce  the  equations 
to  the  form  [3  a]  and  show  that  the  equations  [la]  are  solvable. 

The  theorem  which  has  been  proved  is  of  great  importance 
when  applied  to  special  cases,  even  for  elementary  investigations. 

If,  for  example,  the  equation  /(#,  y)  =  0  is  given,  then  it  is 
taught  in  the  differential  calculus  how  we  can  find  the  derivative 
of  y  considered  as  a  function  of  x. 


164  THEORY  OF  MAXIMA  AND  MINIMA 

If  we  assume  that  the  variables  x  and  y  are  limited  to  a  special 
realm  where  the  two  derivatives  with  respect  to  x  and  y  do  not 
vanish  and  therefore  the  curve  f(x,  y)=§  has  no  singular  points, 
and  if  the  equation  is  satisfied  by  the  system  (#0,  y0),  we  may 
write  x  =  #0  4-  f ,  y  =  yQ  4-  17.  We  have  then  f(xQ  +  f ,  y0  +  17)  =  0, 
and  we  may  prove  with  the  aid  of  the  theorem  in  §  118  that  77  is 
a  continuous  function  of  f  and  has  a  first  derivative.  Not  before 
this  has  been  done  have  we  a  right  to  differentiate  and  proceed 
according  to  the  ordinary  rules  of  the  differential  calculus. 

MISCELLANEOUS  PROBLEMS 

1.  Show  that  the  problem  of  determining  the  extremes  of  the  function 
f(x,  y)  may  be  reduced  to  the  determination  of  the  upper  and  lower  limits 
of  this  function  under  the  condition  that  a;2  +  y2  =  r2.    (Stolz,  Wiener  Ber., 
Vol.  C,  p.  1167.) 

2.  Find  the  shortest  distance  from  the  point  P(xlt  y^,  zx)  to  the  plane 

Ax  +  By  +  Cz  +  D  =  0.  A  ar,  +  By,  +  Cz.  +  D 

Answer.    .  

•\A42  +  B2  +  C2 

3.  Find  the  points  on  a  given  sphere  which  are  the  farthest  from  and 
nearest  to  a*  given  plane  which  does  not  intersect  the  sphere.    (Pappus.) 

4.  Find  the  triangle  of  maximum  area  whose  vertices  F1?  F2,  and  Vs 
describe  respectively  three  given  plane  curves  Clt  C2,  and  Cs.    When  the 
three  curves  reduce  to  the  same  ellipse,  show  that  there  are  an  infinity  of 
triangles  of  maximum  area  (a  case  of  improper  maximum). 

5.  Find  the  ellipse  of  least  area  that  may  be  drawn  through  the  three 
vertices  of  a  triangle. 

6.  Find  the  ellipsoid  of  least  volume  which  may  be  drawn  through  the 
four  vertices  of  a  tetrahedron. 

7.  In  a  triangle  of  greatest  or  least  area  circumscribed  about  a  curve, 
the  points  of  contact  are  the  mid-points  of  the  sides. 

8.  Among  the  triangles  whose  vertices  are  situated  respectively  upon 
three  given  straight  lines  in  space,  which  is  the  one  whose  perimeter  gives 
a  maximum  or  minimum?    Also  determine  the  triangle  of  maximum  or 
minimum  area. 

Answer.  In  the  first  case  the  bisectrices  of  the  triangle  are  respectively 
normal  to  the  straight  lines  described  by  the  vertices ;  in  the  second  case 
the  altitudes  of  the  triangle  are  perpendicular  to  these  lines. 


SPECIAL  CASES 


165 


9.  Upon  a  fixed  surface  find  a  point  P  such  that  the  sum  of  the  squares 
of  its  distances  from  n  fixed  points  Alt  A2,---,  An  is  a  maximum  or  a 
minimum. 

Answer.  If  the  tangent  plane  at  P  is  taken  as  the  a^-plane  and  the  normal 
to  this  plane  at  P  as  the  z-axis,  the  center  of  mean  distances  M,  say,  of  the 
points  A  lie  upon  the  z-axis.  It  follows  that  the  points  P  are  the  feet  of 
the  normals  which  may  be  drawn  to  the  surface  from  M. 

10.  Show  that  the  semi-axes  of  a  central  section  of  a  quadric 
A^x"-  +  A2y2  +  Asz2  +  2  B^yz  +  2  Bnzx  +  2  Bsxy  +  1  =  0 
are  the  roots  r2  of  the  equation 


B9, 
B2, 

I, 


B2, 

Bv 


m, 


=  0, 


where  the  section  is  made  by  the  plane 

Ix  +  my  +  nz  =  0. 

11.  Show  that  the  axes  of  the  quadric  of  the  preceding  example  are 
the  roots  of  the  following  cubic  in  r2  : 


=0. 


12.  z  —  \  v  —  y  is  to  be  a  maximum,  where  ?/3  —  nyx  +  x3  =  0  and  v  —  x  —  y. 
(Hudde,  1658.    See  Descartes,  Geom.,  Vol.  I,  pp.  507-516.) 

13.  The  fundamental  theorem  of  algebra.    Let/(<)  be  an  integral  function 
of  t  with  constant  coefficients.    Write  t  =  x  +  iy,  so  that 

(1)  /(O  =  />(*,  y)  +  iQ  (x,  y)  =  P  +  iQ, 

with  the  identical  relations 


(2) 


dx 


dQ  ,      dP  dQ 

—     and     —  = 

dy  dy  dx 


Form'  the  expression  fj.  ( x,  y)  —  /a  =  P2  +  Q2.  Within  the  circle  of  radius 
r  •=  Vz2  +  y2  the  function  /A  is  everywhere  continuous,  so  that  (§  8)  the 
function  /x  must  reach  its  lower  limit  for  values  of  x  and  y  within  or 
on  the  boundary  of  the  circle.  By  taking  r  sufficiently  large  it  is  seen 
that  the  lower  limit  of  /x  must  be  reached  within  the  circle,  so  that  there 
must  be  a  minimum  value  of  /x.  Show  that  this  minimum  value  is  zero, 
and  consequently  that  there  must  be  some  value  of  t  which  nxakes  f(f)  =  0, 
provided  that  f(f)  is  not  a  constant.  In  particular  the  semi-definite  case 
must  be  considered. 


CHAPTER  VIII 

CERTAIN  FUNDAMENTAL  CONCEPTIONS  IN  THE  THEORY 
OF  ANALYTIC  FUNCTIONS 

I.   ANALYTIC   DEPENDENCE;  ALGEBRAIC   DEPENDENCE 

128.  If  in  the  development  of  the  conception  of  the  analytic 
functions  we  start  with  the  simplest  functions  which  may  be 
expressed  through  arithmetical  operations,  we  come  first  to  the 
rational*  functions  of  one  or  more  variables.  The  conception  of 
these  rational  functions  may  be  easily  extended  by  substituting  in 
their  places  one-valued  functions,  and  first  of  all  those  which  may 
be  expressed  through  arithmetical  operations,  viz.,  sums  of  an 
infinite  number  of  terms  of  which  each  is  a  rational  function,  or 
products  of  an  infinite  number  of  such  functions. 

Such  a  transcendental  function  is,  for  example, 


where  ut  (x)  \i  =  1,  2,  •  -  •  ]  are  rational  functions  of  x.  The 
necessity  at  once  arises  of  developing  the  conditions  of  con 
vergence  of  infinite  series  and  products,  since  such  an  arithmetical 
expression  represents  a  definite  function  only  for  values  of  the 
variable  for  which  it  converges.  Mere  convergence,  however,  is 
not  sufficient  if  we  wish  to  retain  for  the  functions  just  mentioned 
the  properties  which  belong  to  the  rational  and  the  ordinary 
transcendental  functions.  All  such  functions  have  derivatives, 
and  we  shall  restrict  ourselves  once  for  all  to  functions  which 
have  derivatives. 

Furthermore,  the  derived  series  of  the  above  expressions  of  one 
variable  must  converge  uniformly  (cjleiclimassig)  in  the  neighbor 
hood  of  each  definite  value,  and  every  term  of  the  derived  series 

*  See  Hancock,  Elliptic  Functions,  Vol.  I,  pp.  6-9. 
166 


CERTAIN  FUNDAMENTAL  CONCEPTIONS          167 

must  be  continuous  in  the  same  neighborhood.  (Osgood,  Lehrbuch 
der  Functionentheorie,  p.  83.) 

129.  When  we  say  that  a  series  whose  terms  are  functions  of 
one  variable  converges  uniformly,  we  mean  the  following  :  *    It  is 
assumed  that  the  series  in  question  has  a  definite  value  for  x  =  XQ. 
We  consider  all  values  of  x  for  which  x  —  XQ  does  not  exceed  a 
definite  quantity  d.    This  determines  a  fixed  region  for  x,  within 
which  we  shall  suppose  that  the  series  is  convergent.    This  region 
is  known  as  the  region  of  convergence  (Convergenzbezirk).     We 
may,  for  brevity,  put 

Rk(x)    for 

in  the  series  above.  In  order  that  this  series  converge  uniformly, 
it  must  be  possible  after  we  have  assumed  an  arbitrarily  small 
positive  quantity  8,  and  when  a  remainder  Rk(x)  has  been  sepa 
rated  from  the  series,  to  find  a  positive  integer  m  so  that 

|  Rk  (x)  |  ^  8,   where   k  >  m 

for  all  values  of  x  within  the  region  of  convergence.! 

130.  Proceeding  in  this  way  we  may  form  more  complicated 
expressions  ;  for  example,  we  may  let  w  (x)  be  a  sum  of  an  infinite 
number  of  terms  where  each  term  is  a  transcendental  function 
like  v(x)  above,  so  that 


We  may  continue  by  forming  similar  expressions  out  of  the 
transcendental  functions  w(x)y  etc.  It  is  clear  that  if  we  proceed 
in  this  manner,  there  is  no  end  of  such  expressions,  so  that  even 
if  we  limit  ourselves  to  one-valued  functions,  we  do  not  obtain  a 
clear  insight  into  the  possible  kinds  and  forms  of  such  functions. 
It  is  essential  that  all  such  transcendental  functions  have  a 
common  property,  and  we  note  that  if  we  take  a  value  XQ  within 
the  region  of  convergence  in  which  the  series  representing  these 

*  Weierstrass,  Collected  Works,  Vol.  II,  p.  202,  and  Zur  Functionenlehre,  §  1. 
+  See  Dini,  Theorie  der  Funktionen  (page  137  of  the  German  translation  by  Liiroth 
and  Schepp). 


168  THEORY  OF  MAXIMA  AND  MINIMA 

functions  converge  uniformly,  they  may  be  represented  for  all 
the  values  of  x  in  the  neighborhood  of  XQ  as  series  which 
proceed  according  to  positive  integral  powers  of  x  —  XQ  ;  for 
example,  in  the  form 

F(x)  =  F(x  -  £0+  x0)  =  «0+  a^(x  -  xQ)  +  a2(x  - 


where  «0,  av  a2,  •  •  .,  are  definite  functions  of  XQ.  From  this  it 
follows  that  they  may  be  differentiated,  and  a  number  of  other 
properties  are  immediate  consequences. 

131.  We  may  next  extend  the  conception  of  uniform  conver 
gence  to  functions  of  several  variables.  With  Weierstrass  (loc.  cit.) 
consider  the  infinite  series 

v=  oo 

F(xli  x2,  .  .  .,  xn)=^uv(x1,  x2,  .  .  .,  xn) 

v  =  0 

whose  terms  uv  are  functions  of  an  arbitrary  number  of  variable 
quantities  x\,  x2,  •  •  -,  xn.  Such  a  function  converges  uniformly  in 
any  part  (R)  of  its  region  of  convergence  when  with  a  prescribed 
quantity  8  chosen  arbitrarily  small  there  exists  a  positive  integer 
m  such  that  the  absolute  value  of 


for  every  value  of  k  which  is  ^  m  and  for  every  system  of 
values  of  xlt  x2)  •  •  »yxn  which  belongs  to  (R). 

Let  «!,  a2,  •  •  .,  an  be  a  definite  system  of  values  of  the  vari 
ables  xv  x2,  •  •  .,  xn  within  the  region  of  uniform  convergence,  and 
consider  only  the  values  of  xv  x2,  •  •  •,  xn  for  which  x1—a1)  x^—a^ 
•  -  .,  xn  —  an  do  not  exceed  certain  limits  dly  d2,  •  •  •,  dn,  as  in  §  129. 

The  function  may  then  be  represented  through  an  ordinary 
series  which  proceeds  according  to  integral  powers  of  x1  —  alt 
x2—a2,  x3  —  «3,  •  •  .,  xn  —  an,  and  consequently  may  be  differentiated; 
in  short,  it  behaves,  as  Weierstrass  *  was  accustomed  to  express  it, 
like  an  integral  rational  function  in  the  neighborhood  of  a  definite 
position  within  the  interior  of  the  region  of  uniform  convergence. 

*In  this  connection  see  Weierstrass,  Werke,  Vol.  II,  pp.  135  et  seq.,  and  also  Bier- 
mann,  Theorie  der  Analy.  Funktionen,  pp.  429  et  seq. 


CERTAIN  FUNDAMENTAL  CONCEPTIONS         169 

132.  We    may    next    introduce    the    conception    of    analytic 
dependence.    If  we  represent  a  function  which  has  been  formed  as 
indicated  above  by  F(x^  x^t  •  •  .,  xn),  then  F(xlt  x2>  •  '  •>  xn)=  0 
expresses  a  certain  dependence  among  the  variables  xlf  x^,  •  •  • ,  xn ; 
that  is,   among  the    infinite   number   of   systems   of  values   for 
which  the  function  has  a  meaning,  those  only  which  satisfy  this 
equation   are  to  be   considered.    There  exists,   therefore,   among 
xl,.x2)  •  •  -,  xn  a  dependence  of  a  similar  character,  as  in  the  case 
of  algebraic  equations.    If  we  choose  the  quantities  o^,  x%,  -  •  .,  xn 
such  that  the  equations  ^i=0,  F2=  0,  -  •  •,  Fm=  0  exist  where 
m  <  n,  we  have  a  dependence  among  the  quantities  xlf  x2,  •  •  • ,  xn 
defined  in  such  a  way  that  at  all  events  we  can  choose  arbitra 
rily  not  more  than  n  —  m  of  the  variables,  since  the  remaining 
m  variables  are  determined. 

133.  The  conception  of  the  many-valued  functions  is  at  once 
suggested.    Suppose,  for  example,  a  function  of  two  variables  x 
and  y  is  given ;  then  we  may  consider  all  the  systems  of  values 
(x,  y)  in  which  x  has  a  prescribed  value.    For  such  a  value  of  x 
there  may  exist  several  values  of  y.    We  are  to  regard  y  as  a 
function  of  x,  and  this  function  is  a  many-valued  function.    By 
the  introduction  of  one  or  more  auxiliary  variables  it  is  often 
possible  to   express   the    many-valued   functions*   through   one- 
valued  functions,  and  indeed  in  algebraic  form.    The  development 
of  analytic  functions  from  an  arithmetical  or  algebraic  standpoint 
seemed  especially  desirable  to  Weierstrass.    He  wrote  (see  Werke, 
Vol.  II  (Oct.  3,  1875),  p.  235):  "The  more  I  consider  the  under 
lying  principles  of  the  theory  of  functions  —  and  I  do  this  con 
tinually —  the  stronger  am  I  convinced  that  this  theory  must  be 
built  upon  the  foundation  of  algebraic  truths." 

134.  To  illustrate  the  remarks   of  the  preceding  article,  con 
sider  any  analytic  dependence  existing  between,  say,  two  variables 
x  and  y  and  limit  one  of  the  variables  x  to  a  definite  region. 
The  other  variable  must  be  expressed  through  x  and  in  a  fofm 
that  remains  invariably  true  for  all  values  of  x  in  question.    Now, 
if  to  the  one  variable  there  corresponds  a  transcendental  function, 

*  We  might  cite,  for  example,  the  Abelian  transcendents. 


170  THEORY  OF  MAXIMA  AND  MINIMA 

it  does  not  seem  possible  to  express  one  variable  arithmetically 
in  terms  of  the  other.  We  may,  however,  introduce  a  third  aux 
iliary  variable  and  thereby  express  both  of  the  original  variables 
as  one-valued  functions  of  the  third  variable  and  in  such  a  way 
that,  if  we  give  to  this  variable  all  possible  values,  we  have  all 
systems  of  values  of  (x,  y). 

The  simplest  example  is  perhaps  the  one  given  by  the  equation 
z  =  &,  where  z  and  y  are  two  independent  variables.  It  is  not 
possible  to  express  the  dependence  between  x  and  y  in  an  arith 
metical  form ;  that  is,  one  in  which  transcen dentals  do  not  appear. 
But  if  we  introduce  a  third  variable  t,  and  write  x  =  ett  we  have 

z  —  e^,  so  that  y  —  — ^-  • 

Thus  x  and  y  are  expressed  as  one-valued  functions  of  tt  and 
such  that  for  one  value  of  x  there  is  invariably  one  value  of  t 
and  of  y.  Poincare*  proved  that  if  x  and  y  are  connected  by  an 
algebraic  equation,  then  all  systems  of  values  (xt  y)  may  be 
expressed  in  the  form  just  indicated.  He  also  showed  that  if 
any  analytic  dependence  exists  between  x  and  y,  it  is  always 
possible  to  represent  x  and  y  as  one-valued  functions  of  a  third 
variable.  An  example  in  point  is  the  expression  of  the  integrals 
of  linear  differential  equations  through  the  Fuchsian  functions. 
However,  he  did  not  show  in  this  latter  case  that  all  the  points 
of  the  region  in  question  were  thus  expressed  through  t.  On 
the  contrary  it  seems  that  there  exists  an  infinite  number  of 
isolated  points  which  can  be  reached  only  when  t  tends  toward 
certain  limits. 

For  example,  in  the  differential  equation  of  the  hypergeometric 
series,  we  should  have  to  exclude  in  such  a  representation  the  real 
values  of  x  from  + 1  to  +00.  (See  the  Paris  Thesis  of  Goursat.) 

In  this  manner  the  study  of  many-valued  functions  may  be 
reduced  to  the  study  of  one-valued  functions.  However,  it  is  not 

*See  Bulletin  de  la  Socitte  mathtmatique  de  France,  Vol.  XI  (1883).  See  also 
lectures  II  and  III,  delivered  in  the  Cambridge  Colloquium,  by  Professor  Osgood 
(Bulletin  of  the  Amer.  Math,  Society,  1898) ;  and  the  Problemes  mathfrnatiques  of 
Professor  Hilbert  in  the  Comptes  rendus  of  the  Congress  of  Mathematicians,  Paris, 
1900.  In  his  treatment  of  Algebraic  Functions  of  Two  Variables,  Professor  Picard  has 
done  valuable  work  in  this  connection. 


CERTAIN  FUNDAMENTAL  CONCEPTIONS          171 

a  simple  task,  for  if  we  wish  in  reality  to  make  this  representation 
even  in  the  case  of  a  linear  differential  equation,  we  encounter 
many  technical  difficulties.  Nevertheless,  it  is  essential  to  prove 
that  there  exist  such  representations. 

Weierstrass  asserted  (May,  1884)  that  he  believed  the  follow 
ing  theorem  existed  in  the  theory  of  functions:  It  is  always 
possible,  where  an  analytic  dependence  exists,  to  express  this 
dependence  in  a  one-valued  form  which  remains  invariably  true. 

135.  We  may  next  introduce  the  folio  whig  theorem,  which  is 
extensively  used,  particularly  in  the  calculus  of  variations  : 

Suppose  that  between  the  variables  xv  x2,  •  •  •,  xn  we  have  .m 
equations  given  which  may  be  represented  in  the  form  of  power- 
series,  and  let  these  be 


-  «„)  +X  = 


where  ^1,  X2>  .  .  .,  X,rt  are  also  power-series  of  xl  —  av  •  •  •, 
xn  —  an,  but  of  such  a  nature  that  each  term  in  them  is  of  a  higher 
dimension  than  the  first. 

TJie  equations  will  be  satisfied  for  n\  =  alt  •  •  .,  xn=  an.  We 
propose  the  problem  of  determining  all  systems  of  values 
(&i>  fyy  •  •  •)  xn)  which  lie  in  the  neighborhood  of  (alf  a2,  •  >  •,  «„) 
and  which  satisfy  the  m  equations  above  ;  that  is,  among  the 
systems  of  values  for  which  \xl—  a^  •  •  •,  \xn—  an\  are  smaller  than 
a  fixed  limit  p,  determine  those  which  satisfy  the  m  equations. 

The  quantity  p  is  subject  to  the  condition  only  of  being  suffi 
ciently  small.  To  solve  this  problem  we  consider  the  system  of 
linear  equations  to  which  the  given  equations  reduce  when  we 

Xl=o.x,=o...,X.=o. 

Through  these  linear  equations  m  of  the  differences  xl  —  av 
x^—a^-",  xm—  am  may  be  expressed  in  terms  of  the  n  —  m 


172  THEORY  OF  MAXIMA  AND  MINIMA 

remaining,  if  the  determinants  of  the  mth  order  which  may  be 
formed  out  of  the  m  rows  of  the  c's  are  not  all  zero. 

If,  say, 

0, 


we  have  (§117) 


x2-  a2  = 


By  means  of  these  equations  we  may  represent  xl—  av  x2—«2, 

•  •  -,  xm—  am  as  power-series  in  the  remaining  n  —  m  differences, 
the  formal  procedure  being  as  follows  : 

We  write  Xi=  0,  •  •  •,  X^=  °>  an(^  tnus  obtain  for  x1  —  av 

•  •  •  ,  xm  —  am  expressions  which  represent  the  first  approximations. 
These  are  substituted  in  Xj>  *  *  '»XL»  an(^  ^n  tne  resulting  ex 
pressions   only  terms  of   the   second    dimension   are   considered. 
These  terms  added  to  the  terms  of  the  first  approximations  respec 
tively    constitute   the    second    approximations.    Continuing    this 
process  we  may  represent  the  required  expressions  to  any  degree 
of  exactness  desired. 

We  obtain  the  same  results  if  we  express  m  of  the  quantities 
xl—  av  x2—  &2,  •  •  .,  xn—  an  through  power-series  in  terms  of  the 
remaining  n  —  m  quantities  with  indeterminate  coefficients.  These 
coefficients  may  be  determined  without  difficulty. 

As  just  shown,  these  power-series  are  convergent  as  soon  as 
the  differences  x  —  a  which  enter  into  them  do  not  exceed  cer 
tain  limits,  and,  furthermore,  these  power-series  satisfy  the  given 
equations. 

136.  The  problem  of  the  preceding  article  may  be  solved  in 
the  following  more  symmetric  manner,  in  which  none  of  the  vari 
ables  is  given  preference  over  the  others  (see  Lagrange,  Th£orie 
des  Fonctions,  Vol.  II,  §  58). 


CERTAIN  FUNDAMENTAL  CONCEPTIONS 


173 


Besides  the  equations  given  above  we  introduce  others  which 
are  likewise  expressed  in  power-series  : 


'1,1, 

cl,  2,           '  '  *>    cl,  7/i> 
C2,2,            '••>    c2,/n, 

cl,  wi+1,           '  *  *,    ^1,  n 

ii 

7/1,2,                        *>        771,771, 
I,      C/7l+l,2,      '    *    *,      C77l+l,  J7l» 

'7/1,7/1  +  !,          *  "  *,    'm,  » 

CJ»+1,  771+1,      *    *    *»     C77l+l,  71 

<» 

C»,2.           '•••    CnfW, 

'71,  TO+I,          *  *  *,    'n,  n 

where  ^,  £3,  •  •  •,  £„_,„,  are  ?i  —  m  new  or  auxiliary  variables. 

The   quantities   c   are   arbitrarily   chosen,   in   such   a   manner, 
however,  that  the  determinant 


0. 


Proceeding  as  in  §  117  we  write  the  quantities  ^  equal  to 
zero,  and  we  thus  have  a  system  of  n  linear  equations  through 
which  we  can  express  the  n  differences  x1  —  alf  •  •  .,  xn  —  an 
through  t1)t2)"-)tn_m)  say, 

xv  —  av  =  ev  ^  4-  eVi2tz  H h  eVi1l_mtn_m  +X'V 

(^  =  1,2,...,  ?i). 

With  the  help  of  these  equations  we  can  express  xl  —  alt  •  •  ., 
^»  —  «„  as  power-series  in  tlt  •  •  •,  ^_OT. 

To  do  this  we  again  write  ^1  =  0,  anc^  have  only  terms  of 
the  first  dimension.  We  write  the  first  approximations  that  have 
been  thus  obtained  in  ]J£'  and  by  retaining  the  terms  of  the 
second  dimension  derive  the  second  approximations,  etc. 

It  follows  directly  from  the  above  that  these  power-series  in  t 
formally  satisfy  the  given  equation ;  that  they  possess  a  certain 
common  region  of  convergence  if  we  give  certain  fixed  limits  to 
KI|>  |^|>  •  '  '>Ki-m  5  that  they  consequently  in  reality  satisfy  the 
equations  ;  and,  finally,  that  all  the  systems  of  values  (xly  %2,  •  •  -,  xn) 
which  lie  in  the  neighborhood  of  (alf  a2,  •  .  .,  an)  and  which 
satisfy  the  proposed  equations  are  obtained  in  this  way. 


174  THEORY  OF  MAXIMA  AND  MINIMA 

137.  Suppose  that  between  two  variables  there  exists  an  ana 
lytic  relation  which  is  expressed  in  the  form 


where  P  denotes  simply  a  power-series  and  where  XQ,  ?/0  is  a 
definite  pair  of  values  of  the  variables. 

In  the  neighborhood  of  (XQ,  yQ)  there  is  an  infinite  number  of 
systems  of  values  which  satisfy  the  equation.  The  collectivity  of 
these  pairs  of  values  (x,  y)  is  called  an  analytic  structure,  or  con 
figuration  (Gebild),  in  the  realm  (G-ebiet)  of  the  quantities  (x,  y). 

We  may  next  make  an  application  of  the  theorem  of  the  pre 
ceding  article.  It  follows  that,  if  between  n  quantities  xl}  x2, 
•  •  -,  xn  there  exist  m  equations  in  the  form  of  power-series, 
then  the  differences  ^  —  al}  •  •  •,  xm  —  am  may  be  expressed 
through  power-series  of  the  n  —  m  remaining  variables.  Weier- 
strass  said  :  "  Through  the  m  equations  a  structure  of  the  (n  —  m)  th 
kind  in  the  realm  of  the  n  quantities  x±,  x%,  -  -  •,  xn  is  defined." 

As  in  the  case  of  two  variables,  we  may  proceed  in  a  similar 
manner  with  several  variables,  among  which  an  analytic  depend 
ence  exists.  Let  this  connection  be  of  such  a  nature  that  m  (<n) 
of  the  variables  are  in  general  determined  through  the  remaining 
n  —  m.  If,  then,  (alt  a2,  -  -  -  ,  an)  represents  a  definite  system  of 
values  of  the  variables,  there  exist  m  equations  of  the  form 

P(xl  -  alt  x2  -  aa,  •  •  .,  xn  -  an)  =  0 

which  are  to  be  satisfied  for  x1  =  a1}  x2=  az,  •  >  -}  xn=  an.  In  the 
neighborhood  of  the  position  (a^,  a2,  •  •  •,  an)  there  are,  then,  an 
infinite  number  of  other  systems  of  values  (x^  x2,  >  •  •,  xn)  which 
satisfy  the  same  m  equations.  These  define  an  analytic  structure* 
in  the  realm  of  the  quantities  xlf  x2,  -  -  •,  xn. 

A  fundamental  theorem  in  the  theory  of  functions  of  the 
complex  variable  is  that  these  structures  may  be  continued  over 
their  boundaries.  The  power-series  above  constitutes  an  element 
of  a  complete  structure  (§97). 

*  Weierstrass,  Werke,  Vol.  II,  p.  236.  It  may  be  remarked  that  Minkowski  in  his 
Geometric  der  Zahlen  advances  similar  ideas  at  considerable  length.  See  in  particu 
lar  §  19  of  his  work  just  mentioned. 


CERTAIN  FUNDAMENTAL  CONCEPTIONS         175 

138.  Analytical  structures,  as  above  defined,  may  be  represented 
in  a  different  manner.  If  the  equation  connecting  x  and  y  begins 
with  terms  of  the  first  dimension,  we  may,  on  the  one  hand,  either 
express  y  —  y0  through  P(x-xQ)  or  x-x0  through  P(y-y0); 
or,  on  the  other  hand,  if  the  coefficient  of  either  x  —  XQ  or 
y  —  ?/0  is  equal  to  zero,  it  is  possible  to  express  only  y  —  y^ 
or  only  x  —  XQ  as  integral  power-series  of  x  —  XQ  or  y  —  yQ.  In 
order  that  this  distinction  may  not  be  necessary,  we  introduce 
a  function  t  which  begins  with  terms  of  the  first  dimension  in 
x  —  XQ,  y  —  yQ  (see  §  136);  we  may  then  always  express  the  two 
quantities  x,  y  as  power-series  of  t.  Through  the  introduction  of 
such  a  quantity  t  it  is  made  possible  to  include  within  certain 
limits  all  the  systems  of  values  (x  —  xQt  y  —  yQ)  which  satisfy 
this  equation.  These  values  must  firstly  satisfy  the  given  equa 
tion,  and  secondly  they  must  afford  all  the  systems  of  values 
which  satisfy  it  within  these  limits. 

These  considerations  may  be  extended  at  once  to  equations  in 
several  variables.  If  we  have  a  certain  number  of  equations  in 
xl  —  alf  -x%  —  a2,  •  •  .,  xn  —  an,  and  if  we  limit  these  equations 
to  terms  of  the  first  dimension,  we  have  linear  homogeneous 
equations  of  the  first  dimension,  the  number  of  which  we  assume 
to  be  m  (<  n). 

If  we  can  express  m  of  the  quantities  xl  —  alt  x2  —  a2,  •  •  ., 
xn  —  an  through  the  remaining  n  —  m,  it  is  always  possible  so  to 
derive  n  power-series  of  n  —  m  quantities  tlt  t2)  •  -  •,  tn  _  m  that 
they,  substituted  for  xl}  -x^,  •  •  .,  xw  firstly  satisfy  the  given  equa 
tions,  and  secondly,  if  we  give  to  tlf  t2)  •••,  tn_m  all  possible 
values,  they  offer  all  the  systems  of  values  (xly  x2,  •  •  .,  xn)  which 
satisfy  those  equations,  when  certain  limits  are  fixed  for  the  abso 
lute  values  of  %  —  alt  x2  —  a%  •  •  •,  xn  —  an  ;  or,  also  secondly,  that 
with  indefinitely  small  values  of  the  t's  they  afford  all  the  systems 
of  values  of  the  quantities  xl}  x2)  •  •  •,  xn  which  lie  indefinitely 
near  the  position  (al}  a2,  •  •  •,  an)(see  again  §  136). 

Take  n  power-series  ^(t),  ^(O*  •  •  •>  ^n(0  and  write  xl  =  ^1(t), 
xz  =  3*2  W> '  "i  xn  ='®n(t)'>  tnen  through  these  equations  a  struc 
ture  of  the  first  kind  (Stufe)  in  the  realm  of  the  n  quantities  x 


176  THEORY  OF  MAXIMA  AND  MINIMA 

is  defined  ;  in  a  similar  manner  a  structure  of  the  second  kind  is 
defined  through  the  equations 


In  general,  if  we  take  %  power-series  in  tv  t2,  •  •  •,  tn_m  and 
write  these  equal  to  xlf  x2>  •  •  •,  a;w,  the  collectivity  of  the  sys 
tems  of  values  (xv  xz,  •  •  •,  a?w)  offered  through  these  equations 
constitute  a  structure  of  the  (n  —  m)th  kind  in  the  realm  of  the 
quantities  xv  x2,  •  •  •,  xn. 

We  shall  in  the  sequel  limit  the  discussion  of  the  general 
analytic  dependence  to  the  cases  where  this  dependence  is 
expressed  through  algebraic  equations  and  to  the  structures 
which  result  from  such  equations,  viz.,  the  algebraic  structures. 

II.    ALGEBRAIC  STRUCTURES  IN  TWO  VARIABLES 

139.  Let  F(xt  y)  be  an  integral  algebraic  function  of  x  and  y 
which  does  not  contain  repeated  factors,  so  that  F(x,  y)  has  no 

common   factor   with   either  —  -  or  ---    Further   suppose   that 

dx          dy 

F(x,  y)  is  not  divisible  by  any  integral  function  in  which  appears 
only  one  of  the  variables  x  or  y.  The  system  of  values  x,  y  which 
satisfy  the  equation  F(xt  y}=  0  form  the  algebraic  structure  that 
is  defined  through  this  equation. 

If  a;0,  y0  is  a  pair  of  values  such  that  F(xQt  y0)  =  0,  we  may 
develop  the  equation  F(xQt  yQ)  in  powers  of  x  —  XQ  and  y  —  y^  in 
the  form  (cf.  Stolz,  loc.  cit.,  p.  177) 


[1]        G(f,  1,)  =  dF(xQ,  2/0)  +       d*F(xQ9  y,} 


where  for  brevity  we  put  x  —  XQ  =  %  ,  y  —  y^—  rj,  and  where  dnF(xQ, 
is  the  homogeneous  function  of  the  nth  degree  in  f  ,  77,  viz., 

r=n  /Aj\          QW  TJT 

[2]  d*F(xQ,  y,}  =  V  (  v  }  r  ~  V- 


CERTAIN  FUNDAMENTAL  CONCEPTIONS          177 

cF         cF 

If  -  —  and  7r—  do  not  both  vanish,  the  position  (or  point)  XQ,  y  is 

ex          ex 

0  0 

said  to  be  regular  or  simple.  But  if  they  both  vanish  for  x  =  XQ, 
y  =  yQ,  and  if  for  the  same  position  all  the  partial  derivatives  of 
the  2d,  3d,  •  •  -,  (k  —  l)st  order  of  F(x,  y)  with  respect  to  x  and  y 
vanish,  while  those  of  the  &th  order  are  not  all  zero,  the  position 
XQ,  y0  is  called  a  singular  position,  and,  specifically,  a  &-ple 
singularity.  In  such  a  case  the  left-hand  side  of  equation  [1] 
begins  with  terms  of  the  kih  order  with  respect  to  £  and  77. 

In  the  following  treatment  not  only  the  integer  k  plays  an  im 
portant  role  but  also  the  smallest  exponent  of  the  terms  that  are 
free  from  77,  as  also  the  smallest  exponent  of  the  terms  that  are  free 
from  £,  on  the  left-hand  side  of  [1].  If  we  denote  the  first  by  p 
and  the  second  by  q,  the  equation  [1]  may  be  written  in  the  form 

[3]        F^+bys+ri) 

=  e  {«  +  f  /(?)}  +  1?*  {b  +  r,g  (T?)}  +  frHfc  T,)  =  0. 


Here  /(f)  denotes  an  integral  function  of  £  and  g(rj)  an  integral 
function  of  T)  ;  a  and  I  are  constants  different  from  zero,  viz., 

1  cpF  I  cqF 

~~~~          ~~' 


140.  Developments  of  the  algebraic  function  y  in  the  neighbor 
hood  of  a  regular  position.   It  may  be  shown  *  that  if  on  the  position 

7*  77*  / 

x  =  XQ,  y  =  y$  the  expression  —  —  does  not  vanish  I  so  that,  say,  q  =1 

cF\  dy  \ 

and  b  =  -  —  )  ,  there  is  one  and  only  one  convergent  series  in  integral 

i/o  / 
positive  powers  of  £  which  vanishes  with  £  and  which  substituted 

for  rj  in  [1]  identically  satisfies  [1], 

We  may  suppose  that  this  series  begins  with  f  ^,  so  that 

[4]  ,=B_ff*  +  <i(+1f»+i+.... 

We  have   also  to   consider  in   the  sequel  fractional   positive 

i 
powers  of  x  —  xQ  =  j;',  that  is,  powers,  say,  f  **,  where  /*=£!.   A  series 

*  See  Pierpont,  Vol.  I,  p.  288;  or  Goursat,  Coitrs  D'  Analyse,  Vol.  I,  chap.  iii. 


178  THEORY  OF  MAXIMA  AND  MINIMA 

of  this  kind  is  convergent  if  there  is  a  positive  quantity  R  such 

that  the  series  for  all  values  of  \^  <R  is  convergent,  and  that 
is  for  all  values  of  \%\<R*. 

If  the  series  is  convergent  for  one  of  the  /-t  values  of  the  /xth 
root  of  £,  it  is  evidently  convergent  for  all  the  other  /-i  —  1  values 
of  f  .  Accordingly,  to  each  of  the  values  of  f  whose  absolute  value 
is  smaller  than  R*  there  correspond  p  different  values  of  the  series. 

If,  for  example,  we  denote  a  definite  one  of  the  values  of  f,  for 

i 

example,  the  principal  one,  by  f  **,  the  others  are  expressed  through 

the  product  yf  **,  where  j  is  any  of  the  ftth  roots  of  unity. 
Hence  a  series 


may,  corresponding  to  the  different  values  of  jy  appear  in  the  p  —  I 
other  forms  = 


The  theorem  stated  at  the  beginning  of  this  article  may  be 

O  J7I 

generalized  :    If  on  the  position  x  =  a?0,  y  =  yQ  the  expression  — 

9 
does  not  vanish,  there  is  one  and  only  one  convergent  power-series 

in  positive  integral  or  fractional  powers  of  f  which  vanishes  with 
f  and  which  written  for  77  in  the  equation  [1]  identically  satisfies 
it,  viz.,  the  series  [4].  For  if  besides  the  series  [4]  a  series  [5] 
with  /-&>!  satisfied  [1],  then  the  equation 

i 
[6]  F(xQ  +  p,  2/o  +  >?)  =  0,     where  t  =  fr, 

would  be  satisfied  by  two  series  which  have  no  constant  term  and 
in  which  77  is  expressed  in  integral  powers  of  t.  This,  by  the  previ 
ous  theorem,  is  impossible,  because  in  [6]  the  term  in  TJ  really 

O  TJT 

appears,  and  in  fact  multiplied  by  the  coefficient  -  — 

dF  dF         y® 

141.  Suppose  next  that  —  =  0,  but  that  —  ¥=  0,  so  that  p  =  1 

fyo  3xo 

and  q>l.    Then  from  what  we  have  just  seen  it  follows  that  the 


CERTAIN  FUNDAMENTAL  CONCEPTIONS          179 

equation  [3]  may  be  solved  through  only  convergent  series  in 
which  f  is  expressed  in  powers  of  77  hi  the  form 


[7]  £  =  -  -^  +  dtf  +  <  +  •  -  -  =  Q(n),  say, 


where  q>l  and  df  =£  0  ;    in  other  words,  there  exists  a  positive 
quantity  S  such  that  if  !  rj  \  <  S,  we  have  the  identity 

[8] 

Write  f  in  the  form 


and  note  that 

[Q-,(rj)]^  =(  1  —  — !  rjs  H-  •  •  •  }q  =  1  —  ^  77*  -|-  terms  of  higher  order. 
V         b 

If  then  we  put 

[9]  *  =     -fL^lV'/J  '  £ 

by  reverting  this  series  we  have 
[10]  t.,.+^i+.... 

and  from  this  it  is  seen  that  a  positive  quantity  K  may  be  so 
determined  that  for  all  values  of  t  such  that  1 1  \  <  K  the  above 
power-series  in  t  converges.  This  power-series  when  written  for  rj 
in  the  equation  [9]  identically  satisfies  it. 

If,  further,  we  raise  the  equation   [9]   to  the  qth  power  and 

multiply  it  by ,  we  have 

€t> 

b  b  b 

tq  = rnqQ-i('n}= ^  -f  dgif]q +*+...=  Q(rf)  above  ; 

a  a  a 

and  this  equation  will  be  an  identical  one  if  for  ??  we  write  the 
power-series  P(t).  The  same  is  true  of  equation  [8] ;  that  is,  we 
have  the  identity  ,  -, 


for  all  values  t  for  which  the  series  P(t)  is  convergent. 


180  THEORY  OF  MAXIMA  AND  MINIMA 

If  we  denote  the  radius  of  convergence  of  the  series  P(t)  by  R 
and  put  £  = ^,  where  t  is  any  one  of  the  ^-values  of  the  qth 

a 

root  of  —  -  f ,  we  have  the  following  theorem : 


V 


a 


?,  so  that  the  series 


tii]     >^-?~+ijj(jiFjS*1 +•'•• 

exist,  j  denoting  any  of  the  qth  roots  of  unity,  then  this  expression 
written  for  rj  causes  the  function  G(£,  TJ)  to  vanish  identically. 

Furthermore,  there  is  only  one  such  convergent  series  in  inte 
gral  or  fractional  positive  powers  of  f,  without  constant  term, 
which  when  substituted  for  rj  in  equation  [1]  causes  that  equation 
to  vanish  identically. 

For  if  there  were  another  such  series  in  integral  positive  powers 

i 
of  £**,  say, 

then  in  the  manner  given  above  we  could  express  ^,  and  conse- 

i 
quently  also  f ,  through  a  power  series  in  ^  which  identically  satisfied 

[1] ;  but  besides  the  series  [7]  there  exists  no  such  series,  and  conse 
quently  there  is  no  such  series  as  [12]  which  is  different  from  [11]. 

III.    METHOD  OF  FINDING  ALL   SERIES  FOR  y  WHICH 
BELONG  TO  A  /C-PLY  SINGULAR  POSITION* 

142.  In  equation  [1]  let  dF(xQ)  y0),  dzF(xQ,  y0), 
be  zero,  so  that  this  equation  becomes 

[13]    G(f,,)  =  I 


where  N  is  the  dimension  of  F(x,  y)  with  respect  to  x  and  y. 

*  Besides  Stolz,  p.  182,  see  also  Puiseux,  Journ.  de  Math.,  1st  Series,  Vol.  XV, 
p.  365;  Picard,  Traitt  etc.,  Vol.  I,  p.  392;  Hermite's  preface  to  Appell  et  Goursat, 
Fonctions  Alg&briques  etc. ;  Konigsberger,  Elliptische  Functionen,  Vol.  I,  p.  187 
et  seq. ;  etc. 


CERTAIN  FUNDAMENTAL  CONCEPTIONS          181 

There  is,  consequently,  a  &-ple  singularity  at  x0,  y^  and  we  shall 
next  show  that  we  may  derive  all  those  convergent  series  without 
constant  term  which  proceed  in  integral  or  fractional  powers  of  f 
and  which  when  substituted  for  77  in  [13]  identically  satisfy  it,  if 
we  can  derive  corresponding  series  for  any  simple,  double,  up  to 
(k  —  l)-ple  position  of  any  algebraic  structure.  In  other  words,  the 
problem  of  deriving  these  series  for  a  k-ple  singularity  is  made  to 
depend  upon  the  derivation  of  such  series  for  a  position  that  is 
less  than  &-ple. 

If  for  T)  in  the  homogeneous  function*  of  the  nth  dimension 

**(^jrj»  *.(&'*) 

(it  being  supposed  not  identically  zero)  we  write  the  series  [12] 
and  arrange  in  ascending  powers  of  f  ,  then  if  X  =  /i,  this  expres 
sion  begins  at  least  with  f  M,  and  exactly  with  this  term  if  <£>„(!,  CA) 
does  not  vanish.  If  X  ^  /*,  this  expression  begins  with  f  n  only 
when  this  term  in  reality  appears  in  <£„(£,  77)  ;  otherwise  with 
a  term  of  higher  or  lower  order  than  f  w  according  as  X  >  IJL 
or  X  <  fM. 

If  in  [13]  we  next  decompose  the  lowest  differential 


into  its  real  or  complex  linear  factors,  we  have 
[14]  <^.(^)=IIK^ 


r-l 


where  l\  +  &2  +  •  •  •  4-  &/  =  k  and  where  one  of  the  two  coefficients 
ar,  /3r  may  be  zero. 

Assuming  first  that  X  =  /u,  if  in  the  above  expression  we  write 


we  see  at  once  that  for  at  least  one  value  of  r  we  must  have 


*  The  method  given  by  Weierstrass,  Werke,  Vol.  IV,  pp.  19  et  seq.,  is  essentially 
the  same  as  that  found  here  :  see  also  Stolz,  loc.  cit. 


182  THEORY  OF  MAXIMA  AND  MINIMA 


For  if  this  were  not  the  case,  then  £(f,  CA£  +  •  •  •)  would  begin 
with  f  *  instead  of  vanishing  identically. 

If,  next,  X>^)  one  of  the  quantities  j3v  /32,  •  •  •,  ^l  must  be 
zero;  and  if  X</i,  then  one  of  the  quantities  a1?  a2,  •  •  •,  aL  must 

A 

vanish.  For  if  they  were  all  different  from  zero,  then  6r(f  ,  cAf  **+  •  •  •) 

k\ 

begins  with  f  ^  . 

If  a  series  of  the  form  [12],  where  X  =  JJL,  satisfies  the  equation 

[13],  we  shall  have,  if  in  [12]  we  write  ?;  =  (CA  -f  TJ^  £,  a  relation 

I  2 

between  ^  and  f,  viz.,  ??!=  cA+1fA  +  cA  +  2fA  H  ----  . 

The  expression  G(%,  (CA  +  ^I)!)  contains  the  factor  p,  which 
may  be  neglected,  so  that  -^  satisfies  the  equation 


If  in  the  series  [12]  (when  \> /JL)  we  write  r?  =  77^,  we  find  that 
the  equation  -, 

is  satisfied  by  the  series 


If  a  series  for  ?;  where  X</i  satisfies  [13],  we  revert  the  process 
and  make  the  substitution  f  =  rj^r 

143.  In  giving  the  practical  method  of  determining  the  series 
for  r;  which  satisfies  [13]  we  must  make  a  distinction  between 
two  cases :  The  function  <&k  (|,  rj)  either  may  contain  different 
linear  factors  to  their  respective  powers  or  it  is  the  &th  power  of 
one  single  such  factor. 

First  case.    Among  the  quantities  alt  az,  •  •  •,  cci  there  must  be 

at  least  one  which  is  not  zero.    For  each  aL  which  does  not  vanish 

/•? 
we  put  —  =  c®  and  make  in  [13]  the  substitution 

*i 

[15]  *; 

We  may  then  write 


CERTAIN  FUNDAMENTAL  CONCEPTIONS          183 

where  Gl  is  an  integral  function  in  £  and  ^  which  vanishes  for 
=  0  and  ?/  =  0.    If,  for  example,  i  =  1,  we  have 


From  this  it  is  evident  that  the  position  f  =  0,  1^=  0  in  the 
structure  £(f,  i^)  is  at  most  a  &rple  singularity  and  conse 
quently  less  than  a  &-ple,  so  that  the  problem  may  be  regarded 
as  solved,  since,  by  hypothesis,  when  A^<  k  we  have  supposed  that 
we  may  derive  all  power-series  which  satisfy  £(f,  77^  =  0.  For  77, 
through  the  formula  17  =  (e^  +  i^)  f  ,  we  have  series  arranged  in 
integral  or  fractional  positive  powers  of  f  which  substituted  in 
G(%,r})  cause  this  expression  to  vanish  identically.  Besides  these 
series  there  are  no  other  such  series  for  77  which  begin  with  the 
term  c^f. 

If  in  [15]  we  let  r  take  all  the  values  where  ar=£  0,  we  have 
in  this  way  all  those  series  for  77,  where  X  ^  /*,  which  satisfy  the 
equation  G(f,  T?)=  0.  Among  the  quantities  a1?  a2,..-,ai  there 
may  be  one,  for  example  av  which  is  zero.  If  we  consider  rj  and  f 
interchanged  and  then  make  in  [13]  the  substitution  f  =  T?^,  we 
may  derive  all  series  which  proceed  according  to  integral  or  frac 
tional  positive  powers  of  77  with  constant  term  zero  and  which 
when  written  for  f  in  the  equation  £({?,  77)=  0  identically  satisfy 


it,  and  whose  initial  term  is  ^77  x,  where 

By  reverting  each  of  these  series  we  may  express  77  as  series 
in  terms  of  f  which  satisfy  [13],  where  X<^. 

Further,  we  have  all  such  series.  For  if  [13]  was  solved  by 
writing  for  77  a  series  [12],  then  we  also  satisfy  [13]  by  writing 

for  f  a  series  in  integral  positive  powers  of  77*  whose  initial  term 

contains  77^,  where  —  is  an  improper  fraction. 
X 


184  THEORY  OF  MAXIMA  AND  MINIMA 

Second  case.    Let   <£&(£  ,  rj)  =  (ccrj  —  /3f  )*  and  suppose  first  that 
a  =£  0.    We  make  in  [13]  the  substitution 


and  have,  after  division  by  f*,  the  new  equation 
[17]     ^%)  =         ^ 


If  for  this  equation  the  position  f  =  0,  ^  =  0  is  less  than 
a  &-ple  singularity  the  problem  is  by  hypothesis  solved,  or  if  it 
remains  a  &-ple  singularity  and  if  the  polynomial  of  the  terms  of 
the  &th  order  in  f  and  TJI  may  be  decomposed  into  different  linear 
factors,  we  may  proceed  as  in  the  first  case.  It  may  happen,  how 
ever,  that  the  position  f  =  0,  r}1  =  0  is  a  &-ple  singularity  whose 
terms  again  form  the  Jcih  power  of  a  linear  expression  in  f  and 

??!  which  must  necessarily  be  j-(ar)\  ~  £if)fc- 

K  \ 

If,  further,  we  write  in  [17]  ??2  instead  of  rjv  where  rj2  is  defined 
by  the  equation 


the  expression  will  be  divisible  by  f*,  so  that  we  may  write 


(f 


where  G,(fc  %)=  +  f  J5T,(f,  ,,), 

^2(?>  ^2)  being  an  integral  function  of  f  and  ?;2. 

Noting  (-i)  and  (ii)  it  is  seen  that  if  there  is  for  77  a  series 
of  the  form 

[18]  ,  =  J|  +  4p  +  c/+t+..?) 

then  for  f  =  0  the  quantity  ??2  introduced  above  must  be  zero, 
and  T72  must  belong  to  those  series  that  vanish  with  £  and  which 
are  obtained  from  the  equation  6r2  (£,  rj2)  =  0. 


CEKTAIX  FUNDAMENTAL  CONCEPTIONS          185 

This  equation  may  be  solved  as  above  for  rj2  if  the  position 
|  =  0,  7;2  =  0  for  the  structure  (r2(f,  7/2)=  0  is  less  than  a  &-ple 
singularity  or  if  it  is  a  &-ple  singularity  in  which  the  terms  of 
the  kth  order  do  not  constitute  the  &th  power  of  a  linear  func 
tion  of  f  and  7?2.  We  further  have  all  series,  proceeding  according 
to  powers  of  f  without  constant  term,  which  when  substituted  in 
[13]  satisfy  it,  if  we  solve  the  equation 


with  respect  to  ?72  in  all  possible  ways  through  power-series  in  f 
without  constant  term  and  substitute  these  series  for  7?2  in  the 
expression  (cf.  (i)  and  (ii)) 


But  if  the  position  f  =  0,  ??2  =  0  is  a  &-ple  singularity  in  the 
structure  £2(f,  ??2)=0,  and  if  the  terms  of  the  kih  order  form 
the  kth  power  of  a  linear  expression  in  f,  ?;2,  which  must  have 

the  form  —  (arj2  —  /3.2f  )A',  we  must  write  rj3  instead  of  rj2,  where  ??3 

tv  I 

is  denned  by  //32         \»  ..... 

7?2  =  (-^-h773jf,  (tit) 

and  proceed  in  a  similar  manner  as  above. 

Continuing  in  this  manner  it  is  evident  that  if  a  =£  0  we  may 
derive  all  power-series  in  f  without  constant  term  which  written 
for  77  in  the  equation  [13]  identically  satisfy  it,  if  through  a 
series  of  transformations 


we  may  from  the  given  equation  G  (f  ,  77)  =  0  derive  an  equation 
Gh(£>  77/<)=^  whose  left-hand  side  does  not  begin  with  the  kih 
power  of  a  linear  expression  in  f  and  rjh. 

We  must  finally  come  to  such  an  equation  if  F(x,  y)  and  - 

cy 

have  no  divisor  in  common.  For,  since  the  factor  f  *  appears  with 
each  of  the  substitutions  [19],  it  is  easily  shown  that  the  integer 


186  THEOKY  OF  MAXIMA  AND  MINIMA 

li  in  [19]  cannot  pass  a  fixed  limit.  For  if  F  is  of  the  n\h  degree 
in  y,  we  may  always  find  two  integral  functions  U  and  V  in  x  and  y 
where  U  is  at  most  of  the  (n  —  l)st  degree  in  y  and  V  at  most  of  the 
(n  —  2)d  degree  in  y  such  that  there  exists  the  identical  relation 

C  Tfl 

[20]  rJ-(a,y)  +  0J^=.D(*), 

where  D(x)  is  an  integral  function  in  x. 
Furthermore,  since 


it  is  seen  that 


We  also  note  from  the  formula 
J^(aj,  y  +  v}  =  F( 

if  we  make  the  substitution  x  =  x0  +  f  ,  ?/  =  y0  4-  77,  since 


that 


Expanding  the  left-hand  side  of.  this  expression,  it  is  seen  that 


It  follows  that  after  the  substitution  of 

X  =  XQ  +  £,  y  =  y0  +  7j,     where  from  [19] 

[21]  r,  =         + 


CERTAIN  FUNDAMENTAL  CONCEPTIONS          187 

the  left-hand  side  of  [20]  is  seen  to  be  divisible  by  f^'-D.    But 
on  the  right-hand  side  D(xQ  -f  f)  is  of  the  same  degree  d,  say,  in  f 

as  D(x)  is  in  x.    It  follows  that  h  (k  —  1)  ^  d  or  h  ^  -  —  -•- 

K  —  JL 

If,  secondly,  a  =  0,  or  <1>A.  (f  ,  77)  =  (—  /3f  )*,  it  is  seen  that  through 
a  corresponding  change  of  the  method  given  above,  all  series 
which  proceed  according  to  powers  of  77  without  constant  term 
may  be  found  which  when  written  for  f  in  the  equation  [13] 
identically  satisfy  this  equation.  Through  reversion  of  these  series 
we  derive  series  in  powers  of  f  without  constant  term  which  satisfy 
the  equation  [13]  with  respect  to  77,  and  in  fact  all  such  series. 

144.  The  following  theorem  is  proved  by  Stolz  (Math.  Ann., 
Vol.  VIII,  p.  438)  :  If  XQ,  y^  is  a  position  of  the  structure  F(x,  y)=Q 
and  if  this  equation  is  brought  through  the  substitution  x  =  XQ  -f  f  , 
y  =  y0  -f-  77  to  the  form  [3]  above,  viz., 

[3]       F(.cQ  +  f,  #)  +  *?)  =  (p(a 


then  the  collectivity  of  the  convergent  series  hi  integral  positive 

i  A 

powers  of  £  or  f  M,  viz.,  cxf  **  -h  •  •  -  ,  which  vanish  with   £,  and 

when  written  for  77  in  the  equation  [13]  satisfy  it,  are  charac 
terized  through  ^  _-^ 

2>  =  ?>  2,x=*- 

In  these  expressions  /u  is  the  smallest  of  the  roots  of  f  which 

are  contained  to  an  integral  power  in  each  term  of  a  series  in 

i 
question,  and  X  is  the  least  exponent  of  f*4  in  this  series.    This  is 

illustrated  in  the  example  of  the  next  section. 

145.  The  above  theorem  offers  a  check  for  the  determination 
of  all  the  series  which  belong  to  a  singular  position  of  a  function, 
as  is  illustrated  in  the  following  example. 

Example.    For  the  algebraic  structure  denned  through  the  equation 

4  xzy3  -  9  x*y~  +  2  XGI/  -  21  xf  +  8  y1  -  10  a:10  =  0  (i) 

the  point  x  =  0,  y  =  0  is  a  5-ple  singularity.  The  terms  of  the  fifth 
order  in  (/)  are  4  xzyz  and  consequently  may  be  decomposed  into  the 
factors  x  and  y. 


188  THEORY  OF  MAXIMA  AND  MINIMA 

Corresponding  to  the  factor  y,  write  in  (i)  y  —  xyr    The  result  of  the 
substitution  is,  after  division  by  x5, 

4  y}  -  9  xy*  +  2  x\  -  10  a*  -  21  x*y*  +  8  x*y}  =  0.  («) 

The  point  x  =  0,  ^  =  0  is  a  triple  singularity  for  this  structure,  the  terms 
of  the  third  order  being 

4  y»  -  9  xyf  +  2  afy  =  ^(4  ^  -  x)  (^  -  2  x).  (m) 

Corresponding  to  the  first  factor,  write  in  (ii)  yl  =  xyz  and  divide  the 
resulting  equation  by  xs.    We  then  have 

2  y2  -  10  x2  -  9  #2  +  4  y  «  -  21  afy  «  +  8  x«yj  =  0, 
where  x  =  0,  y2  =  0  is  a  simple  point.    From  this  equation  we  have 


We  thus  have  as  a  solution  of  (i) 

y  =  xyl  =  x2y%  =  5  x4  +  terms  of  a  higher  order.  (fy) 

Corresponding  to  the  second  factor  in  (Hi)  write  in  (ii)  yl  =  x(%  +  y2) 
and  divide  the  result  by  x3.    We  then  have 

-  |.va  -  10  x2  -  6  yl  +  4  3/|  +  .  .  •  =  0, 
and  from  this  we  have          ?/2  =  —  4^-  x2  +  -  •  •  . 

It  follows  that  (i)  is  satisfied  by  the  series 


Corresponding  to  the  third  factor  of  (m),  write  in  (zi)  ?/x  =  z(2  +  y2), 
and  dividing  the  result  by  xs  we  have 


From  this  it  follows  that  yz  =  f  x2-  +  •  •  -  ;   and  the  corresponding  value 

°f  y  iS                                      y  =  2^2+f^+....  (y/) 
Returning  to  (i)  write  a;  =  yxl  so  that  («')  becomes 

4  ar»  +  8  y«  -  21  ^  -  9  yar*  +  2  y8a?«  -  10  a:"^  =  0.  (mi) 


For  this  structure  y  =  0,  xl  =  0  is  a  double  point,  the  terms  of  the  second 

order  being  ,  /— .  ,  /-. 

4  x*  4-  8  y2  =  4  (x^  +  *^V2)(a?j  —  iyv2). 

Corresponding  to  the  factors  of  this  expression  we  make  in   (vii)  the 

substitutions 

(viii) 


CERTAIN  FUNDAMENTAL  CONCEPTIONS          189 
If  then  we  divide  through  by  f,  we  have  the  equations 
±(8*2  -  21^)  *V2  -  2lyxi  +  4  xl  +  •  •  •  =  0. 

From  each  of  these  equations  we  derive  series  which  begin  with  the  same 
term,  viz.,  x»  =  Qy  +  •  •  •,  so  that  we  derive  the  two  series 

±i  V2  +  z)  -  ± 


By  inverting  these  series  we  have  the  series  which  proceed  in  ascending 
powers  of  x$,  viz.,  -  :  — 

y=^T-H=+....  (it)  and  (a;) 

We  have  thus  derived  five  power-series  which  proceed  in  integral  or  frac 
tional  powers  of  x  without  constant  term  which  satisfy  (i),  viz.,  (tV), 
(r),  (i-O,  («),  and  (x). 

It  is  further  seen  that  2/ot  =  l  +  l  +  l  +  2-f2  =  7,  which  is  the  smallest 
exponent  of  the  terms  that  are  free  from  x,  while  2A  =  4  +  2  +  2  +1  +  1  =  10, 
which  is  the  smallest  exponent  of  the  terms  that  are  free  from  y  in  (i) 
(Stolz,  p.  195). 


INDEX 


(The  figures  refer  to  the  pages) 


Abelian  transcendents,  169 

Algebra,  fundamental  theorem  of,  165 

Algebraic  curve  expressed  through 
power  series,  36,  53 

Algebraic  function,  its  development 
in  series,  177;  at  a  singular  point. 
180  et  seq. 

Algebraic  structures,  176  et  seq.,  181, 187 

Ambiguous  case,  the,  iv,  27.  See  Semi- 
definite  form 

Analytic  dependence,  166  et  seq.,  169 

Analytic  function,  73 ;  defined,  138 

Analytic  structure,  74,  174 

Appell,  180 

Area,  maximum  area,  143 

Asymptotic  approach,  138 

Auxiliary  variable,  101,  173 

Baltzer,  2 

Bauer,  107 

Bertrand,  v,  33 

Biermann,  136,  155,  168 

Bocher,  148 

Bohlmann  and  Schepp,  iv 

Bois-Reymond,  Paul  du,  2,  7,  74 

Bolzano,  12,  136 

Borchardt,  107 

Boundary,  136  et  seq.,  156  et  seq. 

Brand,  126  et  seq. 

Burnside,  89,  106 

Calculus  of  variations,  iv,  171 

Cantor,  Geschichte  etc.,  15 

Cartesian  Oval,  133 

Cauchy,  3,  7,  92 

Cavalieri,  27 

Center  of  curvature,  117,  125 

Christoffel,  107 

Complete  differential  quotient,  6 

Contact  of  indefinitely  high  order.  36 

Continuation  of  an  analytic  function. 

174 

Continuous  function,  161,  162 
Convergence,  166,  167,  168,  172,  178 
Cremona,  144 
Curvature,  117 
Cusps,  appearance  of,  30 
Cylinder,  trace  of.  31 


Dantscher,    Victor    von.    v,    36,    69 ; 

method  of,  39,  62  et  seq.,  72 
Definite  form,  19;  necessary  condition. 

21,  49,  50,   51,   64,   68,   82,  83,  91, 

92,  109,  111,  114;  conditions  for,  91. 

103 
Derivative,  existence  of  a,  161,  162. 

163,  166 

Descartes,  iv,  165 
Determinant,  the  sisn  of  the,  25  et  seq.. 

28,  29,  30,  32,  38,  51,  52,  59,  60,  83. 

85  et  seq.,  90.  91.  92,  93,  97,  100. 

107,  111;  orthogonal,  149 
Differentiation,    one-sided,    iv,    7.   11 

et  seq. 

Dini,  12,  136,  167 
Distinctness   as   characteristic  of   an 

extreme,  37,  38,  47,  50 
Double  curve,  54 

Double  point,  101 ;  with  distinct  tan 
gents,  29 ;  isolated,  29,  30,  31 

Element  of  a  complete  structure,  174 
Equation  of  secular  variations,  107 
Euclid,  iii,  15,  135 
Euler,  16,   18,   107;  theorem   of,  for 

homogeneous  functions,  84,  155 
Exceptional  cases  involving  a  squared 

factor,  54,  58,  68,  97 
Existence  of  an  extreme,  proof  of,  135. 

146 

Extraordinary  cases  of  extremes,  iv.  19 
Extraordinary  maxima  or  minima,  1, 

6  et  seq.,  17,  19,  43  et  seq.,  74 
Extreme,  or  extreme  value,  v,  2,  53. 

54;  criteria  for,   4,   6,  26,  92.    See 

Maxima  and  minima 
Extreme  curves,  53.  54 

Failure  of  general  criterion,  55 

Fallacious  conclusions.  See  Incorrect 
ness  of  earlier  theories 

Fermat,  iii,  iv.  15,  132;  method  of 
determining  maximum  and  mini 
mum,  iii 

Form.    See  Definite  form 

Fourier,  iii 

Fourier  series,  74 


191 


192 


THEORY  OF  MAXIMA  AND  MINIMA 


Fractional  powers,  178,  182 

Fuchsian  functions,  171 

Function,  rational,   166;   one-valued, 

166;  many-valued,  169 
Function-element,  138,  174 
Fundamental  theorem  of  algebra,  49 

Gauss,  19, 86, 99 ;  principle  of ,  151  et  seq. 

Genocchi-Peano,  1 

Geometrical  interpretations,  6,  24,  31, 

46,  69,  71,  97,  125 
Geometrical  mechanics,  139 
Geometry  of  numbers,  174 
Gergonne,  19 
Goursat,  6,  28,  27,  29,  31,   126,  170, 

177,  180 
Greatest  value,  1,  48,  94.     See  Upper 

and  lower  limits 

Hachette,  107 

Hadamard,  147,  148,  160 

Hancock,  123,  166 

Hankel,  74 

Harkness,  12 

Hermite,  89,  106,  180 

Hilbert,  170 

Homogeneous  functions,  49,  155 

Homogeneous  quadratic  forms,  82,  85, 
103  et  seq. ;  expressed  as  a  sum  of 
squares,  86,  89,  91 ;  with  subsidiary 
conditions,  114 

Hudde,  165 

Huygens,  16 

Hypergeometric  series,  170 

Improper  maxima  and  minima.    See 

Maxima  and  minima 
Incorrectness  of   earlier  theories,  33 

et  seq.,  52 
Indefinite  form,  19,  49,  50,  51,  64,  68, 

82,  106,  116 

Indeterminate  coefficients,  172 
Inflection,  point  of,  6 
Integral  rational  function,  168 
Isolated  point,  29,  31 

Jacobi,  107 
Jordan,  75 

Konigsberger,  180 
Kronecker,  106 
Rummer,  106,  107 

Lagrange,  iii,  v,  4,  18,  22,  26,  33,  43, 
77,  86,  92,  99,  107,  114,  127,  131, 
148,  172 

Laplace,  iii,  107 

Least  squares,  26 


Least  value,  1,  50,  94.  See  Upper  and 
lower  limits 

Left-hand  differential  quotient,  7,  11 

Legendre,  135 

Leibnitz,  3,  15 

Limitation  expressed  through  an  equa 
tion,  150 

Lipschitz,  2,  74 

Lower  limit,  63, 94, 104, 136.  See  Upper 
limit 

Liiroth.    See  Dim 

Maclaurin,  iii,  3,  4,  15,  22,  77 

Maxima  and  minima  (see  also  Extreme 
value),  one  of  the  most  admirable 
applications  of  fluxions,  iv ;  condi 
tions  for,  iv,  4,  40,  99  ;  inaccuracies 
in,  v ;  maximum  defined,  1 ;  mini 
mum,  1;  ordinary  (see  under  Ordi 
nary  etc.);  extraordinary  (see  under 
Extraordinary  etc.);  proper,  2,  5, 11, 
17,  23,  26,  44,  45,  60,  61,  63,  74,  75; 
improper,  2,  5,  17,  23,  26,  31,  50,  59, 
60,  63,  75,  140  et  seq.,  164;  abso 
lute,  2 ;  relative,  2,  21,  96  et  seq. ; 
criteria  for,  4,  7-12,  40-42,  43  et  seq., 
48,  51,  55,  64,  67,  68,  77,  80,  81,  82, 
92,  100,  102,  115,  116;  geometrical 
interpretation  of ,  6,46, 71 ;  erroneous 
criteria,  33 ;  condition  for  proper 
extremes,  40,  42;  condition  for  im 
proper  extremes,  41,  42,  140  et  seq. ; 
criteria  for  relative  maxima  and 
minima,  115 

Mayer,  iv,  v,  2,  79 

Mechanics,  problems  in,  139,  150 ; 
derivation  of  the  ordinary  equa 
tions  of,  152 

Minimal  surfaces,  123 

Minkowski,  174 

Morley.    See  Harkness 

Neighborhood  of,  in  the,  65,  173 
Newton,  discoverer  of  the  calculus,  iii 

One-sided  differential  quotient,  7,  11 

et  seq. 

Orbits  of  planets,  107 
Order  of  a  curve,  54 
Ordinary  maxima  and  minima,  1  etseq., 

17  et  seq.   See  Maxima  and  minima 
Osculating  circle,  117 
Osgood,  167,  170 

Panton.   See  Burnside 
Pappus,  15,  164 

Pascal,  Exercici  etc.,  11 ;  Bepertorium 
etc.,  130 


IXDEX 


193 


Peano,  iv,  v,  2,  6,  12,  18,  21,  31,  33,  34, 
52,  61,  68,  94 

Pendulum,  150 

Petzval,  107 

Picard,  170,  180 

Pierpont,  3,  7,  15,  34,  37,  177 

Poincare,  170.  180 

Poison,  107 

Polygon.    See  Regular  polygon 

Position,  135  et  seq. 

Power-series,  171  et  seq.,  175.  179 

Proper  maxima  or  minima.  See  Max 
ima  and  minima 

Puiseux,  180 

Quadratic  form,  19 ;  expressed  as  a 
sum  of  squares,  86  et  seq.,  89  ;  ap 
plication  of,  92  et  seq.  See  Homo 
geneous  quadratic  forms 

Radius  of  curvature,  117 

Realm,  135,  174 

Reflection  of  a  ray  of  light,  126  et  seq. 

Refraction  of  a  ray  of  light,  131  et  seq. 

Regiomontanus,  16 

Region  of  convergence,  167,  168 

Regular  function,  73 

Regular  point,  177 

Regular  polygon,  140,  142  et  seq.,  147 

Relative    maxima   and    minima.    See 

Maxima 'and  minima 
Reversion  of  series,  153  et  seq. 
Richelot,  106 

Right-hand  differential  quotient,  7,  11 
Roots  of  unity,  180 

Salmon,  107,  120.  122 

Scheeffer,  v,  19,  27,  35.  36,  39,  46.  48, 

50,  62,  70 

Scheeffer's  method,  37 
Scheeffer's  theorem,  43,  46,  55,  59,  60, 

61,  62,  70,  72 

Scheeffer's  theory.  43  et  seq. 
Schepp.    See  Bohlmann  ;  see  also  Dim 
Secular  variations,  equation  of,  107 
Semi-axes  of  a  central  section,  165 
Semi-definite  case,  iv 
Semi-definite  form,  19,  49.  50,  51,  52, 

64,  65,  68,  70  et  seq.,  82,  83,  92,  93, 

106,  116 

Serret,  v,  33,  104,  106,  136 
Severus,  16 


Shortest  distance  to  a  given  surface, 

101,  123 

Simple  point,  177 
Simpson,  16 
Singular  point,  164,  177,  180  et  seq., 

183,  184 

Sluse,  Ren<§  F.  W.  de,  16 
Smallest  value,  1 
Smith,  Edward,  107 
Spherical  triangle,  135 
Squared  factor.  See  Exceptional  cases 
Stolz,  v,  2,  6,  11,  14.  43,  45,  46.  50,  60. 

70,  79,  91.  100,  136,  155,  164,  180, 

181,  187,  189 
Stolzian  theorems,  39  et  seq.,  55,  58, 

70,  72 

Stolz's  added  theorem,  45,  60 
Structure  of  the  first  kind  etc.,  175 
Sturm's  theorem,  51,  106 
Surfaces  of  second  degree,  107 
Sylvester,  89,  107 
System  of  m  equations,  solution  of, 

171  et  seq. 

Tangent,  parallel  to  z-axis,  6 ;  com 
mon  to  two  curves,  34,  35,  36 

Tangential  plane,  28,  31 

Tartaglia,  16 

Taylor's  development  in  series,  v,  4, 
5,  9,  10,  19,  24,  33,  75,  79,  80,  97, 
152,  159 

Taylor-Lagrange  theorem,  43,  47,  77 

Todhunter,  33 

Transcendental  curves,  36 

Transcendental  functions,  167,  169 

Uniform.    See  Convergence 
Upper  and  lower  limits,  2,  12  et  seq., 
55,  57,  94,  104,  136,  137 

Variations,  calculus  of,  iv,  171 
Von  Dantscher.    See  Dantscher 
Voss,  2 

\Veierstrass,  iv.  73,  79,  86,  107,  138, 

167,  168,  169.  174,  181 
Wilson,  E.  B.,  12 
Wirtinger,  148 

Zajaczkowski,  106 
Zenodorus,  142,  147 


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